I've done it many times in various methods
Short breaks on cut price European flights - I always take smaller kites (Psycho, BOT, Kaos and Obsession) packed down to original package size go in hand luggage or hold luggage
Long haul and I take larger Kites as well so I have a full length kite bag that I have a length of 4 or 5 inch drain pipe inside it (this gives the kites some added protection and I can put all the flying lines, wrist straps, spares and consumables as well as a junior hacksaw - see note - in there as well) it goes in oversize luggage (along with strollers and push chairs or golf bags) first time I did that I thought it would take ages to come through to the baggage area but I've since found that the vast majority of times it arrive first along with all the push chairs.
In the USA I get asked if it's rifles or shotguns in the bag during checks but once they know it's Kites they are good as gold
I've never ever been charged extra for them - mention sports equipment and they don't want to charge - Pretty sure they don't charge for strollers with but I reckoned I'd struggle to convince any check in staff that it was a stroller.
Regarding the hacksaw - I did once forget on the return trip and left it in my everyday rucksack which I had as carry on - cue a very interesting discussion with Miami Airport staff - they kept the blade and I continued with just the hacksaw minus the cutting blade
Adrenaline, Psycho, BOT, Kaos, Obsession , Generation, Club sport (2), Dharma SP Ultralight, Baby Tana, Neutrino, Rev and a couple of duvets