Moving a coop is really two jobs: keeping live birds calm and safe, and getting an often odd-shaped structure apart and onto a truck. Handle the animals first, before you touch a single panel of the coop itself.
Pro tip: Move the coop itself a day or two ahead of the birds if your schedule allows, so it is clean, dry, and reassembled before the hens ever see it, which reduces stress-related egg-laying pauses.
Watch out: Never leave chickens in a closed carrier in a hot car; heat stress can kill a bird in well under an hour on a warm day.