Hey there @Zaffie -- in order to remove your pet from the adventure, first you'll need to go into their adventure, take them to the Entrance Gate in Gateway City, and click "Finish Adventure." Then go to your adventure squad page and click "Remove From Adventure Squad" on the pet that you want to remove.

You'll also have to remove all pets from your current house before moving, which can be done either via your Pixpet Inventory or the Clear House Parts feature located under the Housing tab.

Please let me know if anything I said was confusing or if you have more questions! ^^