A very interesting question.
Some years ago I was in the cat fancy for breeding and showing. Some of us have bought showcats in the whole world, from USA, Hungary, U.K. and others. In the cat world the most breeders ship their cats. You must give references to got one of these cats. A friend of mine was the only person who had bad experiences. She buyed a cat in the USA from a very reputable breeder. The cats was 4 months old. The sad thing was, that the cat was not put on a direct flight and the cat kennel stayed half a day
outside on a windy rainy airport. The airport personal was not able to be carefully with this little cat

. My friend got the cat and after 3 days the little one was very ill. She put him to the vet and the little one died. The breeder was very shocked and shipped my friend an 6 months old cat to replace the first one. My friend mustnīt paid the second one (which had the same quality as the first one) and the shipping costs. And the breeder took the cat on a direct flight with no stops and said never using another flight withs stops or changes to another airplane.
I myself was never shipping a cat, I sold them only in gemany. But I have buyed 2 cats in USA too. The first came with her breeder who is a cat show judge - she was judging a show here in germany. The second one come to the airport where I can pick him up quickly.
But I think I would never do it. I want to meet the breeders personally before buying a cat or dog and want to see the living area.