Rules for flying mounts are found on DMG 50. These rules go into some detail which is covered on this page, as well as creature-specific rules found on the individual pages. The three standard flying mounts are the hippogriff, the griffon, and the pegasus.
If time is not taken to properly mount, the rider will tend to fall in the first round of melee.1)
If not flying, mounts will attack on the 2nd and subsequent rounds after they engage in a melee, just like warhorses. If flying, the flying combat rules must be used, which operates differently than the standard melee.
Note the “Damage” rules on DMG 53 – flying creatures will not be able to continue flying after taking a certain amount of damage.
Normal and max loads for a large number of flying creatures are found on WSG 47.