"I think you overestimate their chances" - Grand Moff Tarkin
I think that far too much advantage is being ceded to the flyers in this thread. Don't get me wrong - it is a very powerful advantage, but it is not - on its own - enough to be decisive in the face of a persistent and adaptive enemy.
As part of a combined arms approach, the intelligence available from the air may prove (or may not prove - intelligence is frequently not decisive) to be highly useful, but as a weapons platform, this is not a decisive advantage on a classical/medieval technology level battlefield to the extent that it can win on its own.
In fact, I will go a little farther: without a combined arms approach, a "flyer" based army will not prevail over a determined ground based foe. The ground based enemy will adapt and win over the course of your world history.
Why?
My belief: Air power is just not powerful enough on its own to win a war, occupy ground and control resources.
If you consider the relative importance of Air Power in WW1, an era in which it was possible to fly MUCH faster than any of these flying creatures can fly, and carry more weight than any of these flying creatures can carry, even then...air power was not decisive in the era. It was very helpful at reconnaissance, but as a weapon, it lacked a number of important features:
1 - it could not occupy and physically control land; (Fantasy: This remains true)
2 - you could dig a hole and put a roof over your head to protect yourself from it, essentially denying it any effectiveness as a weapons platform; (Fantasy - *decisively* true)
3 - it was STILL not all that fast. Now, mind you, this is in comparison to bullets - which were much faster - but even still, a WW1 era plane was really not all that fast in the sense that it could not attack with great surprise and it could be meaningfully tracked by ground weapons capable of bringing it down.
(Fantasy - LESS true, but this is still problematic against a determined enemy who has adapted and evolved anti-air defences of military effectiveness.)
4 - it was a relatively fragile craft not able to withstand much damage before it would fall to the earth. Put another way, to wound an air unit is FAR more likely to result in its death than wounding a ground unit is.
(Fantasy: remains reasonably true)
5 - it was not capable of carrying significant enough weight which could meaningfully accomplish great destruction on the ground (again, this is especially true when compared to contemporary ground based weapons in the era of WW1)
(Fantasy: still equally true)
6 - It was not capable of moving enough men and materiel to be preferred over ground based and sea based transport (Fantasy: This remains true)
7 - An air unit is, economically speaking, far more expensive than infantry or artillery, where it's more or less a case of anyone's son will do.
(Fantasy: MORE true)
In our world, the earliest prophet of Air Power, Giulio Douhet, predicted in the 20's that air power would become all important and the dominant force on the battlefield. He would prove to be right - but most of his contemporaries laughed at him at the time. And - here is the important point – in the mid-20’s they were RIGHT to laugh at him.
It was not until the late 1930's that aircraft engines developed significantly and became powerful enough to carry a heavy enough weight, with a significant enough range, that was fast enough that they could not be easily be brought down from the ground and were themselves powerful enough to be a significant weapons platform.
Even still, it was not until the first Gulf War that air based weapons were accurate enough to be decisive on their own as the pre-eminent stand alone weapons platform of the war.
Ok - that's the real world history lesson. How does this translate to the fantasy world?
1 - Air based forces have superior battlefield intelligence.
They know where the enemy is, barring elaborate subterranean efforts to conceal those facts.
Important advantage, but not decisive on its own. An entrenched defender WILL using elaborate roofing and stone to eliminate your advantage.
2 - Air based forces as air cavalry have superior mobility
Important advantage, but my guess is that they will not be able to do anything of great importance with this advantage standing on its own.
3 - An air based army is going to be hard to conceal when moving and consume VAST amounts of food to move.
This is the tricky part. Anyone can get a few thousand people out on to a field. The trouble starts at dinner time.
An air based army cannot carry enough food to feed its mounts. It’s simple logistics. Flying takes more fuel than walking. That’s true whether your fuel is gasoline or oats. Strategically, your army will have a very limited effective range. What this really means is that it simply is not a stand alone army. And like any army - it crawls on its stomach.
4- As a run and gun guerrilla weapon, it cannot be beat and can't be effectively defended against.
Hey. It's not all bad for air power is it? An air based army should be able to cripple the economy of a dispersed agrarian based foe. At least as deep behind enemy lines as you can reach given your units' effective range. Split up in smaller forces, you can disperse and live off the land. You should be able to burn out and starve your enemy.
This night be the only real killer app to flying pegasi.
Unless he stores his food and is in a big hurry not to sit back and be starved out but seeks to march on your home town for a game of drop the rock.
5 - As a weapon platform, the knight on pegasi or archer on hippogriff is less militarily effective than the archers and pikemen he will be attacking.
This is not heroic combat - this is a battlefield.
The enemy will adapt. They will develop portable anti-air weapons - be it arrows or torsion based launchers which are fast enough and powerful enough and - above all - numerous enough to make the flyer pay a price for coming too close to the enemy. And they will develop shields and pikes formation long enough and thick enough to fend the flyers physical attacks off.
Moreover, when you are talking about arrows, you are talking about a military weapon that is highly inaccurate. A longbowman cannot hit the proverbial cow in the ass with a shovel. What he can do is put a shaft 250-350 yards over yonder. And if you group him with a few thousand like minded buddies, train em for a long time, bunch em up and develop a fire control system so all these ranks in depth can time their arrow launches so the whole barrage hits the target area at once, THEN, YES - you can do a lot of damage and rule the battlefield. (NOTE: We STILL do not know how they actually co-ordinated this kind of arrow attack at Agincourt).
And just HOW, pray, do you do this with a whole lot of flying air cav and wings beating so loud the people up there can't even hear a command? You can't bunch em up to be dense enough to achieve that sort of massed arrow attack like the English longbowmen could to hit the enemy ground army effectively – your wingspan of your mounts is too large. If you try to manage it via layering, you’ll be just as likely to kill your buddy beneath you and you present a target large enough that is simply cannot be missed if it can be reached by a ground based weapon. That means you pull back until you are using gravity assist to hit em. And *that* means you can't hit very much you are aiming at. In order to hit anything with any effectiveness, you will have to be VERY bunched up and dense - far more than I think is practically possible...
But let's say I’m wrong and you can.
Even if you could, it will probably take you enough time to arrange your forces for this sort of arrow launch that the enemy is going to see it coming from well off. If you attack from far enough above that nothing they can fire has enough velocity and momentum to reach you, then you have gravity on your side. The foe is not stupid. They are going to disperse, raise the big ass shields over their heads, dig in and **wait you out**.
Now, all of this changes if your forces are suddenly mounted on dragons. That's like an upgrade to a P-51. But if all we are talking about is Knights on Pegasi?
They'll be pretty. A breathtaking sight. The pride of the nation. But on their own, they are not decisive enough to prevail in the long term.