I think with filming greenscreen you would have to have very stable lighting.

Indeed yes. You have separate lights for the green screen and for the subject, and you generally put a greenfilter on the instrument lighting the greenscreen, and that instrument is usually some kind of large, soft, diffuse 2K, with barndoors/black flags keeping that light from spilling or reflecting on the subject. And the subject is several meters away from the green screen to minimize any green reflections too.