Very carefully. We've moved our poor kitties way too much and they have survived every time. One of them isn't afraid of anything and the other is terrified of his own shadow. Good advice on the valium pills. Our Max (the one who's not afraid of anything) cried for 500 miles - Nonstop. And we drugged him. It didn't work. Sometimes it did, this time it didn't. That is a horrible sound to have to listen to for that long of a trip.
If you have skittish cats, put them in a cat carrier and leave it open so they can get out and explore the vehicle (no, not the moving truck!) get a covered litter pan and leave them food and water out so they can eat when they want too. Our skittish one didn't eat the whole drive (1,000 miles). We took them in the hotel and let them out and he was fine. I honestly wouldn't even let them outside even with a leash. Things happen and the noises are horrible for them, to me it's not worth the risk of losing them. And I wouldn't prolong the trip any longer than necessary. It won't matter if you stop extra times to let them out, they won't be relaxed no matter how many times you stop. We did it in 2 days, stopping only for gas and to eat, which we did when we stopped for the gas.
When you get to your new home, put them in the bathroom with food and water and litter and close the door. While you are moving everything inside they'll be safe that way.
Oh!! One piece of advice. If you let them roam around the car and you are at a rest stop and you get out to stretch your legs (not leaving the area of the car) and you leave the keys in the car and you have power locks, well, lets just say don't leave your keys in the car
with power locks even to step out to stretch those legs. I almost learned the hard way in the middle of nowhere!! Our Max used the drivers side arm rest to jump up onto the dash of the car and just as I was opening the door his foot hit the power lock switch!!!
He locked the car. Keys inside and the dog and 2 cats inside. Oy vey, that was too close.