Finally a member question to which I can speak.
Even before my recent purchase of my TT I traveled with my bird. At first it was my cockatiel and now my wife and I take our pekingese "Eddie" and our African Grey "Gonzo"on every trip. I don't cover her up on the trip as long as she seems ok, she enjoys the scenery & our company. Be sure to have a good travel cage with a solid perch and position her facing on the perch the direction that most interest her.Our last trip was over 7 hrs, next month we will travel about 10 hrs with her.
Inside the TT she stays on the table next to the window. We've only taken her outside once because of the extreme heat. We have a screened room to protect from insects or I probably wouldn't consider it at all.
You might start with a short trip to gage their reaction first. I expect Your birds would probably enjoy the trip and your company very much. If I can help please let me know! ;D