We play musical cages all the time. Most rabbits don't mind so much unless they take offense to their new neighbor. I've had a few bucks charge the wire dividing them and a few does get sprayed by bucks who suddenly find themselves next to a doe they like and want to claim. Other than that mine always curiously explore their new cage and then just start eating or flop somewhere. A chunk of pumpkin at this time of year or a few horse treats can distract more nervous ones but the only ones who have had issues moving were also the ones I butchered for being aggressive toward humans and sometimes violently refusing to breed with any buck or the bucks from those lines being so pushy the does don't want to breed with them. The high strung rabbits get culled and I can move even does with new litters They just worriedly check their nest real quick, find their babies, and once again eat some food, dig a little to rearrange the floor if it's not wire, and flop down in the middle of the cage.
I think change is even a good thing. I can see a big difference in the rabbits I've gotten who sat in the same empty wire cage their whole lives. They really don't know what to do with themselves or how to explore and some become very territorial. My little netherland buck is so used to just sitting around that he doesn't know what else he could try when out of his cage or moved to another cage. He wants to see things but he doesn't know how to go about it so he just sort of bounces up and down while turning his head each direction rotated with flattening down in to a bunny log while he debates what to do next. It is amusing but I feel kind of sorry for him.
On the other hand the rabbits we got at a younger age or raised ourselves and have been shuffled from cage to cage to pen to sometimes the colony with boxes to destroy and treats to beg for have absolutely no problems claiming the whole house or horse stable as their own when let out. They are thrilled to see new surroundings even if it's just a new cage with different digging potential due to changes in style of feeder or going from wire floor to solid and a different hay silo design to figure out.