Will the Draught Chocobo be able to fly?

[Jeykama]

snip

Actually it would be fairly simple if you use established rules:
1) Both riders need flight privileges in the zone (same as wedding ring teleport) otherwise it walks
2) If the driver teleports, the rider automatically goes with them just as if they had crossed a zone line. If the rider teleports, the driver has a short-timed prompt to accept or not - if they do they port together mounted, and if they don't accept, the rider teleports on their own.
3) If the rider feels the need to eject while flying, they can dismount and will automatically eject onto a default flying mount. The only issue with this is that the game never forces you to learn a mount so it would require a mandatory mount of some kind.

[Akuryu]

2) If the driver teleports, the rider automatically goes with them just as if they had crossed a zone line. If the rider teleports, the driver has a short-timed prompt to accept or not - if they do they port together mounted, and if they don't accept, the rider teleports on their own.

What if the driver ports to an area the rider doesn't have access to yet?

[RyuRoots]

What if the driver ports to an area the rider doesn't have access to yet?

Either send the access-less player to the nearest aetheryte or tell the driver that they can't do the teleport because someone doesn't have access like when you try to enter an instance that not everyone has unlocked.