I brought 2 cats from the US with me, but I also brought in 5 strays I saved in Azerbaijan. In both cases, there were no issues. They need a certified document from the Department of Agriculture in the US. Your vet fills it out. Just need to be fixed and up to date on their rabies. The Azerbaijani cats just needed their rabies and a sign-off from the government vet.
I rescue and care for cats here in Tbilisi and will be taking about 8 back to the US when I go again. To get them back in, they just need their rabies up to date. They don't even need to be fixed.
The most important thing to consider when moving with your pet is which airline you fly with because the fees can be outrageous. Azerbaijan Airlines is by far the cheapest--25 EUR per cat. Other airlines cost $200 and up.
I hate to see expats here looking to put their pet off on someone right before they leave, because unless you live in the UK with very strict rules, then taking your pet with you is very easy and inexpensive, especially back to the US. I always offer them my papers, but no one has even checked them.
