Pokémon brought in with their trainers will behave as they always have! Regains (including items/pets brought on arrival) are treated a bit differently from general species and objects imprinted on the setting. You can think of it as the difference between something from "outside" the world (characters, pets, items) compared to something created within the world (imprinted species, scenery, objects). The former can be affected and changed by the world and its rules, but will always retain their original traits, even if some are repressed or nullified.

In other words, pet characters are basically the same as player characters, they just don't have enough influence to change the world on their own. We hope that answers your question!