████████ GLOBAL RULES ████████
The Prehistoric age* - You can only use Grass and Normal-types for the next segment.
The Medieval age* - You can only use Fighting and Fairy-types for the next segment, or in alternative, only Normal-types.
Towards the new future - You have to put a Fairy-type Pokémon (or Pokémon that turns into a Fairy-type in later gens) on your team until the end of the game. It has to use Explosion or any of its variants (or if it can't learn any, then it has to use the strongest attack it currently knows) in each Gym Battle. It's not affected by any other rule.
Sounds sort of like a... - Players of Team Apricorns have to put a Pokémon with type combination of Rock and Ground (either Rock/Ground or Ground/Rock) on their team for the following segment. If you can't, you have to put pure Rock or pure Ground-type Pokémon on your team for the following segment. This Pokémon is unaffected by all other rules.
Time heals all wounds - Creationism is no longer a rule.
Time debt - At the end of your next segment, you can permanently box / release up to 4 Pokémon who battled against the last Gym leader / Kahuna / Captain you faced and obtain 10 Challenge Points for each.
In another time... - Devalue and tenthcompation switched up Clueless and The 12 Monkeys rules.
* = given that these rules restrict you to using different types of Pokémon and can become contradictory if applied during the same segment, you are welcome to follow them during subsequent segment, instead of "as soon as you know about them".
Click on a player's name to see their individual rules.
████████ JANP

Pokémon Sapphire
Badges: 5/8
Rules:
Cavegirl - You have to put at least 1 Ground-type Pokémon on your team for this segment.
Fast Thinking - You must complete your battle's final segment in a maximum of 5 turns per enemy Pokémon you will have to fight. If you don't manage to do this you can try again as many times as you want.
Questionable discovery - You may not use any Steel-, Fairy- or Dark-types.
Myths of the past - You have to put at least 1 Dragon-type Pokémon on your team for this segment.
In the name of progress - You can only use the best version of each available item (for Pokéballs, the only kinds allowed are Premier Balls, Ultra Balls, or any other Pokéball that costs at least 1k $ per piece).
Time capsule - Pick one Pokémon you used in this segment. You can't use it until the final segment and you have to use it in that segment.
████████ QWERTY

Pokémon Fire Red
Badges: 4/8
Rules:
Fast catches - For each area, you may only catch the most common Pokémon you can find in the tall grass of by surfing. No other encounter methods are available unless both grass and water to surf on are unavailable.
Item consequences - If you used any healing item during any previous Gym Battle, you can't use any item in this segment. Otherwise, you have to drop half of your healing items.
The Big Bang - You must use Explosion or its variations at least once during each segment's final battle (if you have no way to have a Pokémon of yours learn Protect, you can ignore this rule).
It's just a ring - You have to put at least one Poison-type Pokémon on your team for this segment.
Back to square one - Consider the number of badges you own: subtract 3 from that number, then consider all the areas of the game that are accessible to you if you still were at that number of badges. You can only use Pokémon you can find in those areas, or their evolutions.
████████ DEVA

Pokémon Fire Red
Badges: 8/8
Rules:
The future is now, old man! - All of your Pokémon must learn all new moves via level-up (or HMs if strictly needed to progress).
The 12 Monkeys - You have to put at least one Rock-type Pokémon on your team for this segment.
Back to square one - Consider the number of badges you own: subtract 3 from that number, then consider all the areas of the game that are accessible to you if you still were at that number of badges. You can only use Pokémon you can find in those areas, or their evolutions.
The Big Bang - You must use Explosion or its variations at least once during each segment's final battle (if you have no way to have a Pokémon of yours learn these moves, you can ignore this rule).
The Butterfly Effect - You have to put at least one Bug-type Pokémon on your team for this segment.
Let the past burn - If you used Fire-type Pokémon in any of the previous segments, you have to beat the next segment with a team made just out of Grass-types. Otherwise, your team can only consist of up to 4 Pokémon.
Courtney - You have to put at least one Fire-type Pokémon on your team for this segment.
Fast thinking - You must complete your battle's final segment in a maximum of 5 turns per enemy Pokémon you will have to fight. If you don't manage to do this you can try again as many times as you want.
Item consequences - If you used any healing item during any previous Gym Battle, you can't use any item in this segment. Otherwise, you have to drop half of your healing items.
████████ ELLE

Pokémon White 2
Badges: 8/8
Rules:
Creationism - Your Pokémon cannot evolve during this segment.
Time flows like water - If you used Water-type Pokémon in any of the previous segments, you can only use Pokémon caught during this segment. Otherwise, you can't use TMs for the rest of the game.
The Wright Brothers - You have to put at least one Flying-type Pokémon on your team for this segment.
The Big Bang - You must use Explosion or its variations at least once during each segment's final battle (if you have no way to have a Pokémon of yours learn these moves, you can ignore this rule).
Team Laventon! - As long as you have a Grass-type Pokémon on your team, all Sinnoh Pokémon are unaffected by Global Rules.
Looper - You have to put at least one Dark-type Pokémon on your team for this segment.
Noir - You can only use White and Black Pokémon during this segment. If you can't, you have to use Gray Pokémon during this segment.
Back to square one - Consider the number of badges you own: subtract 3 from that number, then consider all the areas of the game that are accessible to you if you still were at that number of badges. You can only use Pokémon you can find in those areas, or their evolutions.
A thousands-years old vampire - As long as you have at least 100 000 Poké Dollars, any Pokémon that knows HP-draining moves is unaffected by any of your rules.
████████ CUBE

Pokémon Fire Red
Badges: 4/8
Rules:
The time of peace - Your Pokémon may not use attacking moves of their type, unless they cannot learn any eligible move during the current segment.
The Butterfly Effect - You have to put at least one Bug-type Pokémon on your team for this segment.
The Tomorrow War - You have to put at least one Fighting-type Pokémon on your team for this segment.
Clueless - You have to put at least one Normal-type Pokémon on your team for this segment.
A thousands-years old vampire - As long as you have at least 100 000 Poké Dollars, any Pokémon that knows HP-draining moves is unaffected by any of your rules.
████████ JBSD

Pokémon Crystal
Badges: 8/8
Rules:
Team Oak! - As long as you have a Steel-type Pokémon on your team, all Kanto Pokémon are unaffected by Global Rules.
Bill & Ted's Excellent Adventure - You have to put Fairy-type Pokémon (Or Pokémon that becomes Fairy-type) on your team for this segment.
Time blossoms - If you used Grass-type Pokémon in any of the previous segments, you can only use moves learnt during this segment. Otherwise, you can't return to old locations outside of story or healing reasons for the rest of the game.
Time flows like water - If you used Water-type Pokémon in any of the previous segments, you can only use Pokémon caught during this segment. Otherwise, you can't use TMs for the rest of the game.
The Butterfly Effect - You have to put at least one Normal-type Pokémon on your team for this segment.
Ice Queen - You have to put at least one Ice-type Pokémon on your team for this segment.
Time doesn't fly - You may not use any Flying-, Normal- or Psychic-types anymore.
Fast catches - For each area, you may only catch the most common Pokémon you can find in the tall grass of by surfing. No other encounter methods are available unless both grass and water to surf on are unavailable.
The Wright Brothers - You have to put at least one Flying-type Pokémon on your team for this segment.
████████ MEWTWOLOVER

Pokémon Emerald
Badges: 8/8
Rules:
Running out of time - You have to avoid as many Trainer battles as possible. Also, you can't pick any new items found in the overworld.
Time doesn't fly - You may not use any Flying-, Normal- or Psychic-types.
Super-effective consequences - If you used super-effective move during any previous Gym Battle, you have to use only 1 Normal-type Pokémon solo in this segment. Otherwise, you can only use Pokémon that are super-effective against the next Gym in this segment.
Progressive thinking - You must sell all of the Pokéballs you currently have in your bag before beginning to play your next segment (except the Master Ball).
Bill & Ted's Excellent Adventure - You must put at least one Fairy-type Pokémon in your team (or Pokémon that turns into a Fairy-type in later gens).
Bravely Default - You must use Protect or its variations at least two times during each segment's final battle (if you have no way to have a Pokémon of yours learn Protect, you can ignore this rule).
The Wright Brothers - You have to put at least one Flying-type Pokémon on your team for this segment.
Creationism - Your Pokémon cannot evolve during this segment.
Let the past burn - If you used Fire-type Pokémon in any of the previous segments, you have to beat the next segment with a team made just out of Grass-types. Otherwise, your team can only consist of up to 4 Pokémon.
████████ SANIA

Pokémon Sun or Moon
Badges: 3/8
Rules:
Team Laventon! - As long as you have a Grass-type Pokémon on your team, all Sinnoh Pokémon are unaffected by Global Rules.
Fast thinking - You must complete your battle's final segment in a maximum of 5 turns per enemy Pokémon you will have to fight. If you don't manage to do this you can try again as many times as you want.
Time capsule - Pick one Pokémon you used in this segment. You can't use it until the final segment and you have to use it in that segment.
Genwunners - You can only use Pokémon (or Pokémon that will evolve in such Pokémon) introduced in generations outside of your game's generation.
████████ TENTHCOMP

Pokémon Sword
Badges: 5/8
Rules:
Noir - You can only use White and Black Pokémon during this segment. If you can't, you have to use Gray Pokémon during this segment.
Running out of time - You have to avoid as many Trainer battles as possible. Also, you can't pick any new items found in the overworld.
Arianne - You have to put at least one Water-type Pokémon on your team for this segment.
A thousands-years old vampire - As long as you have at least 100 000 Poké Dollars, any Pokémon that knows HP-draining moves is unaffected by any of your rules.
Clueless - You have to put at least one Normal-type Pokémon on your team for this segment.
Bravely Default - You must use Protect or its variations at least two times during each segment's final battle (if you have no way to have a Pokémon of yours learn Protect, you can ignore this rule).
████████ ALEX

Pokémon Leaf Green
Badges: 8/8
Rules:
The Tomorrow War - You have to put Fighting-type Pokémon on your team for this segment.
Inexperienced - If you play on Switch mode, you have to permanently switch to SET mode. Otherwise, you now have a level cap of the Gym Leader's strongest Pokémon for each segment.
The time of peace - Your Pokémon may not use attacking moves of their type, unless they cannot learn any eligible move during the current segment.
Predicting the future - You have to put a Psychic-type Pokémon on your team for this segment.
Time flows like a water - If you used Water-type Pokémon in any of the previous segments, you can only use Pokémon caught during this segment. Otherwise, you can't use TMs for the rest of the game.
Arianne - You have to put at least one Water-type Pokémon on your team for this segment.
Slower than usual - Your Pokémon cannot evolve by any other method other than level up / friendship. In that case, they can only evolve once their level is equal to their normal evolution level (or the level at which they obtained max friendship) + 5.
A thousands-years old vampire - As long as you have at least 100 000 Poké Dollars, any Pokémon that knows HP-draining moves is unaffected by any of your rules.
The revenge is best served cold - You have to put at least one Posion-type Pokémon on your team for this segment.
████████ JUNO

Pokémon Emerald
Badges: 3/8
Rules:
A thousands-years old vampire - As long as you have at least 100 000 Poké Dollars, any Pokémon that knows HP-draining moves is unaffected by any of your rules.
Fast catches - For each area, you may only catch the most common Pokémon you can find in the tall grass of by surfing. No other encounter methods are available unless both grass and water to surf on are unavailable.
Courtney - You have to put at least one Fire-type Pokémon on your team for this segment.
The future is now, old man! - All of your Pokémon must learn all new moves via level-up (or HMs if strictly needed to progress).
████████ ARC

Pokémon Soul Silver
Badges: 8/8
Rules:
Genwunners - You can only use Pokémon (or Pokémon that will evolve in such Pokémon) introduced in generations outside of your game's generation.
Donnie Darko - You have to put at least one Electric-type Pokémon on your team for this segment.
Creationism - Your Pokémon cannot evolve during this segment.
Inexperienced - If you play on Switch mode, you have to permanently switch to SET mode. Otherwise, you now have a level cap of the Gym Leader's strongest Pokémon for each segment.
Item consequences - If you used any healing item during any previous Gym Battle, you can't use any item in this segment. Otherwise, you have to drop half of your healing items.
Time blossoms - If you used Grass-type Pokémon in any of the previous segments, you can only use moves learnt during this segment. Otherwise, you can't return to old locations outside of story or healing reasons for the rest of the game.
Wait for it... - You must use a two-turn move (including Bide) at least once during your segment's final battle.
Time capsule - Pick one Pokémon you used in this segment. You can't use it until the final segment and you have to use it in that segment.
The revenge is best served cold - You have to put at least one Poison-type Pokémon on your team for this segment.
████████ LEGOLDENWOLF

Pokémon Emerald
Badges: 2/8
Rules:
Fast thinking - You must complete your battle's final segment in a maximum of 5 turns per enemy Pokémon you will have to fight. If you don't manage to do this you can try again as many times as you want.
Back to square one - Consider the number of badges you own: subtract 3 from that number, then consider all the areas of the game that are accessible to you if you still were at that number of badges. You can only use Pokémon you can find in those areas, or their evolutions.
Donnie Darko - You have to put at least one Electric-type Pokémon on your team for this segment.
████████ CELESTE

Pokémon Gold
Badges: 1/8
Rules:
The correct time period - You may only battle one segment's final gym leader / captain / kahuna with Pokémon caught in that segment. Once you get this rule, you can choose during which segment it will apply.
Clueless - You have to put at least one Normal-type Pokémon on your team for this segment.
████████ EXP OF TIME

Pokémon Black 2 (Blaze Black 2 if possible)
Badges: 4/8
Rules:
Slower than usual - Your Pokémon cannot evolve by any other method other than level up / friendship. In that case, they can only evolve once their level is equal to their normal evolution level (or the level at which they obtained max friendship) + 5.
Genwunners - You can only use Pokémon (or Pokémon that will evolve in such Pokémon) introduced in generations outside of your game's generation.
Time flows like a water - If you used Water-type Pokémon in any of the previous segments, you can only use Pokémon caught during this (current) segment for this segment. Otherwise, you can't use TMs for the rest of the game.
Questionable discovery - You may not use any Steel-, Fairy- or Dark-types.
Item consequences - If you used any healing item during any previous Gym Battle, you can't use any item in this segment. Otherwise, you have to drop half of your healing items.
Noir - You can only use White and Black Pokémon during this segment. If you can't, you have to use Gray Pokémon during this segment.
████████ TW BLADE

Pokémon Heart Gold
Badges: 678
Rules:
In the name of progress - You can only use the best version of each available item (for Pokéballs, the only kinds allowed are Premier Balls, Ultra Balls, or any other Pokéball that costs at least 1k $ per piece).
Fast catches - For each area, you may only catch the most common Pokémon you can find in the tall grass of by surfing. No other encounter methods are available unless both grass and water to surf on are unavailable.
Slower than usual - Your Pokémon cannot evolve by any other method other than level up / friendship. In that case, they can only evolve once their level is equal to their normal evolution level (or the level at which they obtained max friendship) + 5.
The correct time period - You may only battle one segment's final gym leader / captain / kahuna with Pokémon caught in that segment. Once you get this rule, you can choose during which segment it will apply.
Noir - You can only use White and Black Pokémon during this segment. If you can't, you have to use Gray Pokémon during this segment.
Super-effective consequences - If you used super-effective move during any previous Gym Battle, you have to use only 1 Normal-type Pokémon solo in this segment, ignoring all other rules that would limit your team somehow. Otherwise, you can only use Pokémon that are super-effective against the next Gym in this segment.
The Tomorrow War - You have to put at least one Fighting-type Pokémon on your team for this segment.
Bravely Default - You must use Protect or its variations at least two times during each segment's final battle (if you have no way to have a Pokémon of yours learn Protect, you can ignore this rule).
████████ GLITCH VIOLET

Pokémon Sapphire
Badges: 0/8
Rules:
The time of peace - Your Pokémon may not use attacking moves of their type, unless they cannot learn any eligible move during the current segment.
████████ DRACO

Pokémon Ruby
Badges: 0/8
Rules:
Cavegirl - You have to put Ground-type Pokémon on your team for this segment.
████████ THE OTHER WOLF

Pokémon Emerald
Badges: 4/8
Rules:
Bill & Ted's Excellent Adventure - You must put at least one Fairy-type Pokémon in your team (or Pokémon that turns into a Fairy-type in later gens) for this segment.
Fast catches - For each area, you may only catch the most common Pokémon you can find in the tall grass of by surfing. No other encounter methods are available unless both grass and water to surf on are unavailable.
Donnie Darko - You must put at least one Electric-type Pokémon in your team for this segment.
Time capsule - Pick one Pokémon you used in the last segment. You can't use it until the final segment and you have to use it in that segment.
Inexperienced - If you play on Switch mode, you have to permanently switch to SET mode. Otherwise, you now have a level cap of the Gym Leader's strongest Pokémon for each segment.
████████ JIN

Pokémon Emerald
Badges: 2/8
Rules:
Donnie Darko - You have to put at least one Electric-type Pokémon on your team for this segment.
Questionable discovery - You may not use any Steel-, Fairy- or Dark-types.
In the name of progress - You can only use the best version of each available item (for Pokéballs, the only kinds allowed are Premier Balls, Ultra Balls, or any other Pokéball that costs at least 1k $ per piece).
████████ KAIZO

Pokémon Black 2
Badges: 0/8
Rules:
Tenet - You have to put at least one Ghost-type Pokémon on your team for this segment. (If none is available, this will apply as soon as a Ghost-type will be available).
████████ APRO

Pokémon X
Badges: 6/8
Rules:
Team Oak! - As long as you have a Steel-type Pokémon on your team, all Kanto Pokémon are unaffected by Global Rules.
The future is now, old man! - All of your Pokémon must learn all new moves via level-up (or HMs if strictly needed to progress).
Donnie Darko - You have to put at least one Electric-type Pokémon on your team for this segment. (If none is available, this will apply as soon as a Electric-type will be available).
Slower than usual - Your Pokémon cannot evolve by any other method other than level up / friendship. In that case, they can only evolve once their level is equal to their normal evolution level (or the level at which they obtained max friendship) + 5.
Fast thinking - You must complete your battle's final segment in a maximum of 5 turns per enemy Pokémon you will have to fight. If you don't manage to do this you can try again as many times as you want.
12 Monkeys - You have to put at least one Rock-type Pokémon on your team for this segment. (If none is available, this will apply as soon as a Rock-type will be available).
Time blossoms - If you used Grass-type Pokémon in any of the previous segments, you can only use moves learnt during this segment. Otherwise, you can't return to old locations outside of story reasons.