So you want to fly your mount in Azeroth? Contrary to what Blizzard intended, it is actually possible, but you will need to be a level 70 hunter funnily enough.
1. Level a hunter to 70
2. Put 1-2 talent points into Animal Handler (BM talent that increases pet %hit/mount speed)
3. Go through an instance portal, and you'll still be mounted. This works for flying mounts so I'm assuming it works just as well for regular ones too.