I moved several times in and out of apartments with Spockie, and even drove us from Massachusetts to Florida over 3 days. I’ve moved both ways – bringing him first and bringing him last.
The best move was the one where I had the keys to the empty, carpeted apt. and went over with my sleeping bag before the furniture was due to arrive the next day. One room was his, (actually it was the only spot without carpeting) so I just brought all of his supplies, his cage, food, toys, and set it up first. I was able to close the door to that room while all the other things came in the next day. He heard the noise, but it wasn’t particularly disturbing since he was tucked into his familiar house.
The cute part was having him in the empty livingroom with me. He raced from one end to the other, doing binkies! He didn’t have any furniture to hide under, or block his run and it was like a great field. In case I was worried that he wouldn’t adjust, he made it very clear that he favored the move.
Even when I would bring him last, I always designated my spot for his house ahead of time, or if it was in the open, I temporarily had him stay in a room with a door. He would inspect the new space, but as long as his stuff was exactly the same within his cage, and I was there, he was fine. A few extra treats that night, or as a distraction at the height of the moving noise never hurts.
By the way, the best moving advice I’ve gotten for humans is to designate one room for order and peace to retreat to. ie, make a bedroom, or a guest room, or a den (bunny’s room?) without ANY boxes, with clear chairs, bed or couch, a light, maybe the TV. It’s where you get away from unpacking to rest quietly without jumping up to unpack "just one more thing."