Pokémon Home has a bit of an odd relationship with LGPE; it's able to store and send Pokémon back to LGPE if, and only if, the most recent game was LGPE. Even Pokémon sent directly from Go to Home can't be sent to LGPE even if they would have been otherwise.
Champions, again, has a bit of an odd relationship with Home. Pokémon sent to Champions are actually just held in stasis in Home, and after returning, their "most recent game" isn't affected by Champions, though they do get a new symbol on their summary screen indicating that they've been to Champions. And I'm not sure if this would prevent a Pokémon from returning or if the "most recent game" mark is the only one that matters.
