As a base point, I think this video of United States Air Force Air Combat Manuevering ( https://www.youtube.com/watch?v=HvDz4MrYXNc ) is a great starting point for grasping the most important part of any combat scenario: splitting the enemy force into more manageable targets.
Now that we have the basis of any strategical victory laid out, we must assess the battlefield. For this, we will use open, unobstructed sky.
If we were to take out the gravitational pull of the planet these birds were on, the direction of anything would be absolutely pointless. Therefor, we know now that the only, and most important resource this battlefield offers is gravity. With this, aerial warfare "high ground" get a different set of bonuses than land warfare "high ground". The avian units can use this resource to build speed faster towards enemies, lob weapons at enemies placed directly under them for extra speed and not having the weapon "arc" with gravity. Basically, be creative as gravity is your main combat modifier.
As for an entire "war campaign", it would be important that any stronghold would have sentries posted 360 degrees around and also be scouting in all directions. Along with this, creating a network of outposts and strongholds would allow for your avian armies to transport weapons and food in a more organized fashion. Imagine these hubs like a web of airports that they are trying to expand out into the enemy army's territory.
Another answer by a user named Hollow suggests using a scorched earth tactic, and that never hurts when trying to dismantle webs of outposts.
It should also be noted that these birds have no communications technology so strategically taking out an outpost without any escapees or messengers living could yield for enemy parties to enter the airspace of the newly claimed fortress completely unaware for a short time. If an outpost doesn't report in a timely manner, a large force could be sent out to investigate.
I hope this is the kind of content you were hoping for, I tried to be diverse in my answer since I wasn't sure if you wanted battle tactics or big picture war strategies.