If you add "in the wind" to the end of the sentence, then they all sound roughly the same to me. Also, they sound more natural to me this way.
If you keep the sentence as it is (i.e., without adding "in the wind"), then "fluttering" sounds the most natural to me, and "flying" the least natural. However, there is not much difference between them, to me. They sound similar to each other.
By the way, there are also other words that work in this sentence, as well, including "blowing", "billowing" and "wafting". As with the original four words you provided, all of these sound better to me if you add "in the wind" at the end.