I moved from Kentucky to South Florida in 2 days so I'm familiar with this.
I called my vet and asked what to do. She gave me a prescription for some mild sedatives and said to fast them the night before the move so they wouldn't have to use the bathroom as much. I fed them each half a pill about a half-hour before getting in the car and they were drugged up by the time we left, barely able to walk actually. Broke my heart to see it but...
They slept pretty much the whole way down to the Florida state line, where I stopped for the night. It was a good 12 hours I'd say with stops and such. And they never left me any surprises. I bought a small litter box with some liners and let them out of their cages once we were in the hotel room. Everything went fine, I fed them and then fasted them again overnight, then repeated the pill process the next morning. It went without a hitch. I think it's less traumatic on them because they're so sleepy they don't realize what's going on.
P.S. They look really stoned when they're on the meds and their little meows are so faint and weak. I freaked but don't worry, they're OK, just high. LOL.