There is currently an event called the Moogle Treasure Trove going on in FFXIV. It ends when Shadowbringers launches, so only about three weeks left.
If you do dungeons to collect their special tomes, then you can get some mounts, minions and other items. But it takes 30 tomes for a mount, and 50 tomes for other cool items, so it takes a long time. Stop by an Itinerant Moogle in the capital cities to see what they offer. After two long days of spamming the low level dungeons with a few level 50 MSQ Dungeons tossed in, I was able to get a mount that I’ve been wanting – The Ixtal Dire Wolf.
And like almost all the mounts, it flies too.