I have a master plan for when we move. I know we're going to move, I just don't don't know where because we haven't bought a house yet. But first we're getting husband a pick-up truck. Then I'm going to stalk CL for a flat bed trailer. When move day comes, I'm putting the birds back into dog crates like when I first got them. Unscrewing the floor of the coop and pulling it up. It's one of those metal shed kits, though I am unsure of how heavy it got after adding wooden window frames and insulation. But before, you could stand inside it, lift up, and move it across the yard.
So in theory, we can lift it, move it, get it onto the flat bed. Take it to the new place, where ever that will be. Shouldn't be more than an hour. The run is the same size, so we can do the same with it. Get it set back up in a new spot, get the birds, move them back in.
As far as moving just birds... I've done it 4 hours repeatedly because I bought them out of state. One time I was rear ended on the interstate, crossing the bridge to home. 1st car of a 5 car pile up. I was cut off, and everyone behind me was going too fast and much too close. Was driving my dad's old station wagon. The impact of the dodge ram pickup behind us bent in the tail gate and knocked the lid loose on the birds. The final truck, a giant Excursion towing a boat, sealed the deal and smooshed everyone again, and the lid got knocked off. State trooper came around asking who had the free ranging chickens in the front car.
LOL.. I did. So, now I never put the birds in the back. I buckle them into the seat belts on the back seat and put suitcases in the back. I was the only one in the pile of cars to not get a ticket. Crazy. That stretch of interstate is weird. Another time during high traffic, I was BACKED into by the car in front of me. Like they didn't know how their brakes worked or something.
I've moved dogs 5,000 miles with an airplane and all. When moving any animal, the most important thing is to keep them calm and hydrated. Dark boxes with good airflow. Treats of wet food, like tomato or watermelon since it can't "spill" and splash around. Use an absorbent bedding to keep the smell down. If you know you'll be staying in hotels, your best bet is a stock trailer for the birds so you can leave them outside and lock them up. Use rachet straps to secure cages inside the trailer.
Moving with chickens is way better than moving with cats. Chickens seem to go with the flow pretty well.