In the fight between Ash and Falkner, his Dodrio was able to "fly" (in quotes) and also use the "move fly" to attak.
He just jump very high, so his fast and very high jump skills, could travel greath distances in little time and also alowed many air moves.
His jump is kind the jump of "the incredible Hulk" of marvel.
Besides, many Pokemons can use the same move with very different ways
for example the "fly" move of dodrio is used different than a Pidgey who just fly litterally, but the result is the same, both for attak and for travel. The way it is executed is not very important.
Beartic can use Aerial ace, but Beartic can't fly, so how is that possible?
In the fight between Ash and Brycen. His beartic used that move, he not fly and cut the oponent with a wing full of wind like a pidgeot, But he jump fast full of wind and cut the enemy with the claw. The way a pidgeot and Beartic execute the Aerial ace is different, but the result is the same.
Same goes for Dóduo, even in the attak and travel, the way may be different, but the result is the same and realistic.