I have flown a few times to pick up a kitten, flow with cats and kittens to shows etc. It was more stressful on me than on the kitten. I found areas where I could take the kitten out, with a walking jacket, while waiting for connecting flights and play with them. Southwest is more allowable to this than United. United squacked at me once in front of the boarding area. Some smaller airports are less picky. If they get on you, just walk to a different area as most airports are large enough to do so. I have never had one poop, pee, or get worked up on a flight. Sometimes they meow a little but a tired kitten is a sleepy kitten so playing is key here.
My second F1 was very young and I flew with her from Chicago to California with no issues. Everyone loved to see her and ask about her, but some old lady that said she looked like a wild animal with a killer instinct and moved away from my seat. I think she was the sweetest cat I have had since so not sure where the "killer instinct" comment came from?!