Honestly, I didn't notice much difference between them and "normal" battles. Still went out of my way to hunt them down for the extra experience just for my flying types, but battle-wise they didn't feel much different. Though I had a Vivillon and Charizard X from the beginning of the game, both of which I adored using, so I may have been a tad biased.
Definitely not incorporated into the story, but at least these gave you a little extra experience if you had at least one flying pokemon to use Fly. I imagine everyone does.. I can't say how many of those are 'battle ready' and kept in the party, but I have always kept mine as an active party member. Just made travel easier.
I'm honestly surprised at how many people don't have at least one flying type at the ready.. I thought that was more common than this, lol.