You can't fly in any but the lightest armour
Birds must be very light to endure staying airborne for very long, and though the largest birds of prey can lift animals as large as goats, it's only for shorter distances. Most avians would only be able to cover their most vital organs with metal, meaning they would not be able to fly into a volley of arrows with the same confidence as a heavily armoured foot soldier.
Furthermore, their large wings would be even harder to armour, and even if they fold them up when diving down on enemy soldiers, they need to spread them afterwards to fly away again, making them vulnerable to crippling wing shots.
The Leonin could counter Avian spear fliers by mixing archers (bow or crossbow) with their shield bearers (who could use spears just as long as the Avians'), allowing the shield bearers to protect the archers while the archers engage the fliers whenever they're out of spear's reach.
The hardest weapon to counter for the Leonins would actually be common rocks, as a rock dropped from 100 metres height will break your arm weather you have a shield or not. In order to avoid these, the Leonin army would have to disperse so each soldier can dodge rocks on their own, and the Leonin army would do well to avoid fighting in open, rocky places. The Leonin might also invent shields with a supporting leg that can take the force of a falling rock, but the added bulk would make them much less fit for use in melee.
Supply trains can't fly
Even if the Avians can fly in battle, they need tents for sleeping, extra clothing, cooking utensils, food rations (to supplement whenever raiding the local countryside won't work), tools for repairing weapons and fletching new arrows, and so on. All these things are probably too heavy to fly around with, so they need to travel overground, not to mention a supply of rocks for whenever they're campaigning in less rocky places. Furthermore, after a few rounds of flying up with heavy rocks, tired Avians would need a place for falling back to rest.
Over time, the Leonin tactics could shift toward surviving the initial onslaught, then charging this camp. The Avians, poorly suited for land-bound combat, have no good way of protecting the camp if they fail to stop Leonin from reaching it.
Mounting casualties encourage non-confrontation
As battles become bloodier for both sides involved, both sides are encouraged to find some target to attack which isn't the other army. Here you can think of the Thirty Years' War: Armies marching around, pillaging the lands for food and plunder as they siege down castles and cities currently held by their opponent. Not a lot of people are killed in battle, more are starved as their food stores are raided over and over, or just killed for being perceived as sympathisers of the opponent. Anyone who has the chance to leave will leave. Hundreds of years later, Leonin parents will still tell their children to behave or the Avians will come, and vice versa.
Make the invention of counter tactics play a part of explaining the ebb and flow of the war. So the Avians start out really proud of their spear fliers, making it the core of their army, but at first confrontations they suffer crushing defeats, allowing Leonins to penetrate far into Avian lands before the latter switch to simple rock dropping out of desperation. This proves super effective, putting the Leonins at full retreat and allows the Avians to retake all lost land and then some. Then this culminates inside Leonin lands as the latter learn to protect themselves and ambush the Avians.
The mountain homeland of the birds is going to be very hard for the Leonins to penetrate, since they're slow to travel through, more sparsely populated (i.e. have less food to forage off unfortunate farmers along the way) and provide the Avians with plenty of safe spots to strike out from. However, no city can survive without adjacent woodlands for fuel, giving the Leonin somewhere to hide in relative safety while laying siege to the cities. Maybe the Leonin have good night vision and the birds don't, allowing the former to advance and attack during the night. And maybe the Avians never built very strong fortifications in their homeland, never having experienced an invasion from land-bound enemies before now and finding walls mostly ineffective against local warlords.