Birds are more problematic than dogs or cats due to temperatures as you mentioned - but the key is the airline company. Ideally, you want the dog or cat with you, but that does not seem to be an option anymore. Like BC already said - you have to work with the airlines. Animal transport companies make a bundle so ideally you want to at least have the pet on your aircraft. The last time I transported a pet I had a customized crate made, got the appropriate paperwork and ok, anesthetized the animal and picked up the groggy cat when I arrived at my destination.