The wave "free-falls" toward the object with exactly the same acceleration as anything else passing by.
I'm not sure what you meant by the last part. The light beams would form a spiral expanding through space. (Like "zooming in" to the spiral, not turning like a screw.) Each photon moves in a straight line, but since the source of the beam is turning, the beam is a curve though space.
Ah you know what - I really wasn't thinking this out.
If waves didn't obey momentum then light being emitted by the very monitor I am looking at now would probably not make it to my eyes, or at least be shifted, since the planet and galaxy are moving rapidly.
I'm not sure what you meant by the last part. The light beams would form a spiral expanding through space. (Like "zooming in" to the spiral, not turning like a screw.) Each photon moves in a straight line, but since the source of the beam is turning, the beam is a curve though space.