My 4yo cat handled the trip pretty well, overall. He usually detests being in a carrier so when I picked him up I thought he would be his typical whiny self, but he was practically mute (I imagine he cried a bit at first but gave up at some point). His crate was clean so I think he held it during the flight.
Once he got here, it took about a week before he fully settled in (eating/drinking/pooping regularly). He was fine during that week, but obviously a bit off his game. He also had to adjust to being an indoor-only kitty now (in NC he got to roam the neighborhood at will), so that took another couple of weeks.
My dog, on the other hand, walked off the plane like nothing had happened, so each animal handles it in their own way, obviously.