Could these warriors make the same kinds of acrobatic flights that airplanes did on real aerial battles? I hope I'm not spoiling the fun, but in order for the creature to sustain flight with an armored person on their back, they would have to be so large the person probably wouldn't be engaging in any melee. There would have to be something preventing the wind resistance from snapping their neck if undergoing a barrel roll or something, and anchoring them firmly enough to somehow aim a weapon would be pretty tricky. I would imagine they would stick mostly to holding on to their steering mechanism, and helping guide the creature. The stakes would be higher with no cockpit, so I would think they'd be more effective guiding the dragon or griffin or not bothering to ride at all. Intelligence of the creature also matters.
How would the military formations be? This depends on how in control of the creatures they riders are. If they are in complete control, it would depend on the capabilities of the creatures. If we're talking teeth and claws, the creatures would probably make the rules of their own melee and the riders could only provide basic commands like go, stay, that way, attack, flee, and any special commands. If the creatures can be equipped with a sword like extension on their face, now we're talking some serious trainable strategy. Flanking would be ideal to bite off wings (see below).
What would be the best tactics for warriors to fight each other? Would long range fighting bow and arrows be sufficiently precise? Would short range joust-like fighting be feasible? I think I've kind of answered this, but I don't see how they could muster the strength to throw a spear or aim a bow (straight back). 160
I would think they would have to use crossbows if anything. The creatures would be encouraged to bite off each others wings as that would result in instant victory. However, the counter-attack would be to grab on to the enemy once your wing is damaged. It would be risky to latch on to another creature since there is no leverage in the air to throw an opponent. So they'd stick to slashing and wing-biting.
Would the catapults from the cities be able to disrupt the battle, or inflict any damage, knowing that the warriors could dodge on all three axes? There may be ways to spray and pray with arrows, use nets, huge ballistic bolts, distracting fireworks, or other weapons depending on the technology of the time. If the creatures fly high enough (which any sensible creature/rider would) the cities would not be able to attack, unless first drawing the creatures in. The only way I can see ground support being involved is if the creatures are attacking something on the ground or caught unaware, otherwise the riders would be using the creatures to attack each other for ownership of the skies whereby they could have the military advantage of dropping stuff or scouting at great distances.
Sorry for being so practical, but to me this kind of fighting seems mostly fantastic and a smart military leader would probably make the battles less epic, and more aimed at military advantage. In any case, we need more information about the world to flesh this out. You have a lot of good directions to go. What about some kind of Geneva convention where they forbid attacking wings at the threat of all creature clans ganging up against violators.
I would stay the heck off, personally =D 46