It may be nice to have it moving as an option in the Tool Window Functions Menu or added into Follow Playback Option because in the Tool Window, Marker highlight is moving according to changes within the Audio Editor.
When naming markers through the Tool Window, user may get confused with the marker selection in the Audio Editor moving on top, and the cursor (as shown in first video)
(because cursor is red and invisible alternating that can be mistaken for snap to marker)
One more question, users generally insert markers and move them with mouse. Have you ever thought of a feature that can move markers using keyboard?