I brought my 2 cats from US to Dubai this past Nov. We used Pet Relocation.com. They are based out of Texas and handled everything, door to door. Our contact is Scotty and he was wonderful to work with. This company works directly with Dubai Kennels & Cattery, again great to work with. Good luck!
I recently relocated from USA to Dubai with two, 9 year old cats. This was the first time they ever flew. They flew KLM, starting in Florida with a stopover in Germany, before they arrived in Dubai, safely, clean, fed and watered. KLM took excellent care of them, including cleaning out crate on stopover.