I've moved house with buns a couple of times, and what I've always done is move the buns first, before the rest of the house. When I've moved them I've tended to restrict their free range space to start with to give them time to get used to the new smells etc, I've taken the buns over to the new place the morning of the move, or a few days before when I was able to stay there with them, and then they don't get in the way with all the hassle of moving day. Then once you start moving in the furniture, keep them in one room, in their cage, preferably a bedroom so it's more quiet, until everything's in and settled. It's not ideal in terms of exercise but it's only one day, two at worst.
That's my approach anyway, and I've done three relatively hassle free bunny moves in the last three years. Good luck!