Wingshooting is a general term for the art of using a shotgun to kill something that is flying. The victim is nearly always a bird, and so the term bird hunting is currently more popular, particularly in America. However, skeet shooting is also often counted as a type of wingshooting, as is drone hunting.
Those who partake claim that the challenge of taking down a prey that can move in three dimensions at high speed makes this more a matter of instinct and art, as compared to two-dimensional ground-based hunting. This may be true, but they are also using a special gun that makes a big cloud of bullets, which seems a bit artless to those of us who have not experienced the sport firsthand.
'Wingshooting' is written however you damn well wish, with a space, hyphen, or nothing at all separating the words, and it can likewise be conjugated with wild abandon; wing shooters like to wing-shoot, but wingshootees dislike being wangshot. 'Wing shooting' is the current favorite in most dictionaries.