The first day
You should better take your new pet home in the morning, and preferably before days off or vacation for you to have time to set the good relations with the kitten before you start working.
Take the kitten in the container during the whole way home. Frightened kitten can quickly rush out of your hand and get into trouble. Put some newspapers on the floor of container and cover it with an old towel. So the floor will be dry if something is wrong with the kitten.
If the kitten soils the corner of the cage and can keep far from that place, maybe you'd like to let everything as it is. It depends on the time of journey, on the mood of kitten, on your skills in relation with kittens and on the power of smell in the car. If the kitten is frightened, if he cries, shrinks, his pupils are wide you should not clean the container because this will only double stress. But if the kitten is quiet then clean the container quickly and drive on.
In fact, you couldn't do much to relax frightened kitten. It has just been taken out from the world he knew very well and set into the noisy moving metal box the place he couldn't even imagine, and also with the man he has never met. So who can accuse him on his fear? As soon as he knows how beautiful his new home is he will be satisfied with the pleased purr. But don't wait such reaction during the first journey by car.
If you want to clean the cage stop the car. And, please, don't do it taking the wheel with one hand and opening cage with the other. Then start cleaning having all the windows and doors closed. If you are driving alone try not to take the kitten out the cage while cleaning it. The kitten can get into some quiet place you don't know about. When it is possible just put your hand into the cage, take the dirty towel out, put the new one and shut the door. Put the dirty towel into one of the plastic bags you have taken. Speak calmly and take everything as a fact. And it will help him to relax.
And now drive on. Having reached home put the kitten into the safe room which was prepared before. Put the cage on the floor and open the door. Sit comfortable, read a magazine. Don't try to make the kitten leave the cage. In some time he'll do it himself. Let him do it. When he does it look after him for some minutes, look how he explores everything around. As soon as he feels himself comfortable take him carefully and put into the toilet box.
This time is not suitable for guests and party connected with new kitten. You must understand his situation. Let him learn to trust his new home. You may invite only one or two friends is some days. Don't let impatient people follow your kitten "just for getting acquainted with him". If the kitten gets away you must respect his decision. With such actions he says "Help me! I'm tired!" The sooner he understands that you will protect his wirld when he needs the sooner he trusts you.