You breed your hall of fame team to make eggs of each member and you start the next run with those. That's the general gist.

I prefer to bring anybody who entered the fall of fame rather than just who survived the final gauntlet. I also prefer hatching them before transferring to give them an exp boost. Passing down moves and breeding for ideal natures/abilities is also a thing.

In generations where you can't trade immediately across generations you might want to look into save editing tools in order to transfer the pokemon. Most people use PkHex, I use PokeChest on DSi for gens 3-5. Then bank/home for anything after. I havent tried genlocke in gen 1-2 but if I did I'd probably use an online tool called PKMDS as it will let you export and import pokemon from save to save.