It costs a fortune, especially these days, to transport a kit. Many drummers just have kits waiting for them at gigs--that's one of the reasons for becoming an endorser of drums. Especially if you are with a large company like Tama, Yamaha, Mapex, DW, and your drumming/band commands good attention, these companies have locations around the world and will be able to supply a kit for the drummer upon arrival. This is one reason why, for example, Tim Alexander left Pork Pie for Tama in the early 1990s. Once Primus got really big and started doing a lot of touring, it became easier for Alexander to have Tama provide him with kits based on their global presence. While Pork Pie makes great drums, they do not have the global presence or financial power of a Tama or Yamaha.
Some drummers do take their kits on tour with them, and use hard shell cases stored in the airlines. I think Steve Smith did this with Journey in the 1980s: Journey was successful enough that Smith could afford to have his kit flown around the world. Again, this can get really pricey, so it is not always a great option. Plus, airlines can lose or damage your equipment. I believe Jeff Tain Watts had a set of original Sabian jazz cymbals he had hand picked destroyed by an airline luggage crew some time ago, for example, so there is always some risk--financial or storage--in transporting your drums over long distances.
I'd prefer to show up to a gig and have a kit waiting. Lugging around drums and setting them up gets really old after doing it for a while.
Yeah, there's some obstacles to overcome. I've only ever flown with a practice pad and sticks, and I got a mean look for the sticks in my carry-on. Modern Drummer had an article about drummers flying once. Often times it's easiest to use an overseas kit instead of transporting one, so it helps if you're an endorser. If it's a sort of music festival that lots of bands are flying to I would imagine a house kit to be available. Some big name drummers in the MD article-- Bill Stewart, Peter Erskine, Jim Riley-- said that they won't take their kit overseas but might take cymbals or a snare along with sticks, drum keys, small things etc. Even then they're really careful. I think Bill Stewart said he has taken his snare on with sticks and other small things in a carry on.