Okay so this isn't the most interesting challenge since there are literally 7 flying type families you can catch before the post game. I'm fairly deep into the run already but the rules are as follows;
1. All Pokemon used must be Flying-type as soon as a Flying-type Pokemon is available.
2. As soon as a Flying-type Pokemon is captured, no Pokemon of other types can be used.
3. That's pretty much it. Flying types aren't plentiful early but these rules are pretty much just a formality because since I'm already 6 badges in.