Instructions
Move kittens only if you must. If your family is moving across country, for example, then you will have to go ahead with the transportation of the kittens. Birth and death are often unexpected occurrences--and this is no different in the cat world. Once you've assessed that you must move the kittens, don't move them away from their mother too soon--and by soon, that means under 8 weeks.
Put the kittens and their mom in a deluxe, ultra-large cat carrier. The mother won't like this. She won't want you to move them. She may hiss and scratch. Gingerly scoop the kittens up, perhaps with your partner, and into the box. Don't worry, mother will follow. She may scratch you on the way in, so wear long sleeves. The kittens are too small to bother you too much. They may yelp slightly. The important point is for them to have a cozy home that mirrors the one they are leaving. After the mother is also in the cat carrier, close the latch.
Move carefully; don't make any sudden movements. Ideally, you won't have to go too far. Once you're in the car, too, make sure the kittens are snug as can be with their mum. Put the box right beside you, ideally, or if you have a passenger, beside them in the back. Someone needs to be able to hold on to the top of the box so it doesn't move around or can catch it in the event of a sudden stop.
Insulate the cat carrier, inside and out. Put a cushion on either side of the carrier. You don't want the kittens to get rattled around. Have a little soft padding inside, but not too much. If you are traveling a long distance, include a small amount of water in a tiny dish, not too much though, as you do not want any spilling. You will be feeding the mother cat once you're stopped and out of the car.
Take care of the kittens first, once you reach your destination. Mother will still be mad at you for having done this to her young. Don't take it personally. Take the cat carrier and put it in the house or apartment and immediately open the latch in the cozy room where the kittens will live. A closet can actually be a very warm, cozy living environment for cats for those first few days of life (with the door open, of course).
How to Move Newborn Kittens
Newborn kittens should not be separated from their mothers for at least 8 weeks. So if you have to move kittens, plan on moving ma along with them. Plan the move carefully, with ample time for driving slowly so as not to disturb the kittens (but not so slowly that you will be in an accident). Remember, you are trying to emulate the comfort of the womb throughout the move. Don't worry, though; these are kittens, not embryos. They don't need to be handled with surgical "kid gloves," just the kind of loving care they deserve.