Yes, I move my Pokémon from Pokémon HOME to future games. A major part of my play style is to begin my Pokémon adventure with my favorite Pokémon rather than just build teams with what I catch throughout the game, which requires me to transfer Pokémon from a previous game. Of course, I do this by hatching Pokémon with good IVs, EV train them with vitamins, transfer them to Pokémon HOME while they are level 1 at the least or level 5 at the most, and transfer them to the new game I'm playing. This allows me to raise my favorite Pokémon with my Starter Pokémon in the new game, leaving me with only the Starter Pokémon to EV train. I typically play Pokémon using two teams, which I refer to as my primary and secondary team. More often than not, my secondary team features the new Pokémon in the new game I'm playing, but not always, as it mainly depends on whether or not I like the Pokémon. In the Virtual Console, I took advantage of a glitch that can guarantee a shiny Ditto with perfect IVs. With that, I always transferred them to newer games using Pokémon Bank and eventually Pokémon HOME. Long before Pokémon HOME existed, I always found ways to transfer older Pokémon to the newer game. In Pokémon Stadium 2, I did this using the N64 box feature, for example. At least to me personally, transferring Pokémon from older games and raising them from a low level along with your Starter is a very fun way to play.
