We left one cat and brought one dog. For us it isn't about the animal's 'emotions'. We believe that when you get a pet, you are responsible for it for life, so it is more about our own personal sense of responsibility.
In the UK, there are many homeless pets in shelters and there are even more in the USA. So by rehoming a pet in the USA or the UK, you are not doing homeless pets any favours. Our cat remained with part of our extended family, we pay her vet bills and we get regular reports on her welfare. The reason we didn't bring her is that she was 14 years old and has very white fur with very pink skin. Not suitable for a sunny climate. Nevertheless, we have maintained our responsibility for her. We have also adopted a local dog and will be adopting more local pets later. We are so glad that we brought our own dog from the UK. She has never been happier or healthier. We had one bout of tick fever and that is it. We get so much pleasure out of sharing in her fun, diving off of piers, chasing Jesus lizards and diving for crabs. We are the one's who benefit emotionally.
Also, just because you bring your pets with you, it doesn't mean that you can't give a good home to some local dogs and cats.