As I’ve suggested before, a fighter that scores a kill is promoted to an Ace (rolls @3).
Air combat lasts a maximum of 3 rounds, perhaps only 2. Aircraft had very limited fuel capacities.
If, at the end of this time, both sides have fighters remaining then no side gets air supremacy. All planes must land with their ground units, and cannot strafe.
Fighters can pull out of dogfights after any round of combat, attackers choosing first. Planes simply land with their own troops. They cannot fire in the ground combat, but can be taken as casualties.
Only rolls of 1 actually destroy an enemy aircraft. Rolls of 2 (or 3 from Aces) only “force down” the rival fighter, which must land as per pulling out of a fight.
If there is any dispute over which plane shot down an enemy fighter and gets the promotion, a “kill” takes precedence over a “down”. Otherwise, if the owning players cannot agree, then nobody gets the credit.
Multiple mixed fighters should roll individual dice in combat; its handy to have a few dice of different colours to facilitate this.