A note regarding importing puppets:
Box 21 is your team and starter, if you don't have it in your team/you chose another starter. Box 22 and beyond is your old box. All held items are removed.
Their levels, ranks and moves are retained. But as Shard of Dreams has introduced a different learnset for the puppets, some moves are now labeled "!" in front of the move name. They're usable ingame, but if you try to bring them to netplay, you'll be rejected and said that the puppet has incompatible skills. This is because it would be illegal to have a move that can't be learned by the puppet.