I was wondering if anyone has experience traveling with transatlantic (Western Europe to Mexico) with a pet bird? From what little research I did I found out that some airlines do seem to allow this, but I was more wondering about comfort for the animal and risks associated with such travel. Specifically the birds are Budgies (these ones: https://en.wikipedia.org/wiki/Budgerigar).
Turkish airlines allow this on board in the cabin:
Only small cats, dogs and birds (goldfinches, budgies and canaries) may travel in-cabin in a compliant airline pet carrier. The maximum carrier size is 16"W x 9"H x 22"L. (23x40x55cm cm) Flexible carriers are suitable as long as they will compress down to 9" and hard carriers are also permitted as long as they meet the maximum measurements.
If your bird fits comfortably in the carrier, they have any necessary vaccinations and you have all the paperwork there should be no risks.