Get a large dog crate and put the litterbox food and box for kittens inside it. This way there is a door to close so you can take them to the new home secure. I would not let the mother cat out again period. She could be lost, attacked by dogs, poisined, shot by neighbor kids, or attacked by the cats that live there now.
Unless she is a breeding queen in a breeding program she will need to be fixed when the kittens are 5-6 weeks old. You can begin to wean the kittens then with soft food.
Other things to worry about. If she hasn't had kittens before and is less than a year and a half she may reject the kittens. There is also the added stress of the move that may cause her to reject as well. You will be faced with feeding, cleaning, and keeping the kittens warm/cool. They will need to be fed every 2 hours around the clock. The kitten formula must be warm to the touch and not heated with the microwave as it can cause burns. After each feeding the kitten needs their tummy gently rubbed and the little bottom rubbed to cause them to pee and potty. Use a moist warm washrag to do this.
To help her de-stress during the trip, bring some feliway spray for her crate. Make sure one of your sweaty work out shirts are with her. Your scent is the best calming agent you can have. Keep the temp in the car as comfortable as possible. If it gets too hot, ziplock some ice into plastic bags and lay them on top of the carrier. The cold will filter down to the queen and kittens.
Good luck and let us know how things turnout.