I just tried it and both host and network client move in sync if the screen size is the same. If not, one of them might move more often but in all cases the scrolling commands are executed on host and client.
And I could not explain why it should be different because host and client use the same code to perform scrolling.
Do you maybe have the Show Local Lyrics option enabled on the client and there is a version of that song on that client, that has different scrolling commands?