This will be the third or fourth time I've linked their
historical origin. Dragoons were cavalry that also wielded guns (sometimes lances) with bayonets. Put Nael van Darnus (with his gunhalberd) or a Musketeer on a chocobo and you'd have a more accurate Dragoon than anything currently playable.
That being said, Square-Enix as a whole has made references to Dragoons riding dragons since FF2. Most notably, Drakengard (also named Drag-On Dragoon in Japanese) focuses all of its gameplay on switching between ground and dragon-mounted combat, and the characters even Jump to their dragons, and deal slight damage when they Jump back down. The difference being they can stay riding their dragon and even guide its flight and attacks, though the Dragon's involvement is entirely voluntary.