Check with both countries Agricultural Department (?) if the species you keep can be exported/imported. Also see if any disease testing and microchips are required for transportation.
Then check with the various airlines what their pet transportation requirements are. Some may allow under seat, some may require in cargo (in a controlled temperature compartment).
I know that parrots are allowed on board in some American Airlines if the Airline Approved Carrier fits underneath the seat. If it doesn't, the bird must be transported via Cargo. This applies to anywhere in the US the bird may be flying to (except for maybe Hawaii). If flying International, the bird *must* be kept in Cargo and would probably require an ID of some sort as well as health certificates. Coming back into the USA usually/may requires a 30-60(?) day quarantine at an approved quarantine facility.
Birds on CITES 1 and 2 may be more difficult, if not impossible, to transport than CITES 3.