A Dave Murphy in this discussion back in 1997 says Séamus Ennis traced an origin to "The Bold Forester". When I looked for lyrics of the Bold Forester, I found a version of Child 73, Lord Thomas and Fair Ellender/Fair Eleanor/Fair Annet; and also a song in the DT, Robin Hood and the Ranger. Neither of those seems to me to be related to either She Moved through the Fair or Out of the Window.

I really wanted to know about the S. Uist Gaelic song, mentioned in the same post. I had the luck recently to come across David Murphy, a traditional singer from Ireland who now lives Scotland, at a singing circle on Zoom. He says he knows versions of "The Bold Forester" but doesn't know of any related Gaelic song. And he doesn't think he was the Dave Murphy who wrote the 1997 comment.