Each story locks out your current PC, giving you a new starter on arrival, and access to a new set of boxes. You play through each DLC's story with an all new roster of Pokémon, each region featuring exclusively Pokémon missing from the base game.
Once you complete the story, you can access all your boxes from anywhere. Similarly, you can return to mainland Paldea, and have access to your original boxes there, at any point.
Starters in Kitakami are Torchic, Mudkip, Treecko, starters in Blueberry Academy are Bulbasaur, Squirtle, Charmander.