It is best to transport kittens in special carriers. It is advisable to put inside something to which the kitten has long been accustomed, which has a familiar smell. It is better to go in a car, as there are a lot of extraneous and frightening noises in public transport for all cats.
When you arrive at the new place, do not rush to get the pet. You can put the carrier on a stable surface, then open the lid and try to pet the kitten, talking to it very affectionately.
It may happen that the animal will hide in a secluded corner (under the couch, behind the closet) out of fear. Do not rush to get him out of there, let him get used to the new surroundings. In the meantime, prepare fresh drinking water, food and a toilet, so that the kitten does not poop in the wrong place.
If there is already another kitten or adult cat in the house, let them into the new room where the kitten lives periodically. Of course, at first they may hiss at each other. It usually takes them a couple of days to get used to the smell.