You could accomplish similar things to a movie-likecutscene with sprites.
In fact, you could even, I dunno, move the camera away from the player, create a massive sprite like the S.S Anne (I don't think that's really controllable with a normal movesprite command though) or several sprites that look like one giant image, off the top of my head, and make it move to create the illusion you've got a proper cutscene going on. Instead of having left,right,up,down animations, replace those with your movie frames, and call the appropriate 'movement' to summon that frame. Delete the sprites when finished.
You could also do tiles and replacing tiles with the new ones.
Lots of things are possible if you have enough patience and are inventive.
edit, because I posted this almost at exact same time as Dizzy and would like to reply to them
A graphic cutscene like the Rayquaza one in Emerald would be extremely difficult to make. Graphic background and object manipulation on this level is the hardest thing you can accomplish as a romhacker.
You need to be proficient in C, you need to know the GBA hardware, you need to know about sprite manipulations(and this is light college-tier math), you need to know GF's object structures, GBA modes and more.
I couldn't do it and I can think of only one person in our community who could pull it off and one other person who could do a similar animation but of worse quality.
I don't think they actually need to do full on sprite manipulations here, I think they only need to fake the effect. Pre-create all the images you need, maybe chop them into tiles, hijack some already existing method of showing images like alter-tile methods, such as Game freak's tile changing scripts, or something like the showmugshot people have but do it in a sequence and have it across the entire screen (or have the entire animation in mugshot sized area like Yellow).
This would be an incredible pain in the ass to do, but hypothetically I think you could do it with freaking scripts, movecamera and tiles, if the only thing you cared about was faking the effect and not actually accomplishing the 'real deal' full quality version.