choose MouseMove Action for the elemet
sym.getComposition().getStage().moveSlide( e.pageX );
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I am not familiar with After Effects, so I can't vcomment on the workflow there.
Edge 2 does have a path option and you can animate an element along that path (see captured)
- to operate along a path you need to set thee transition mode to Motion Path, then alter the x,y (Location) along the Timeline. As captured you can see the path direction (mazelike).
The drawback is there is no easy way to motionpath an elements width along the path as you describe, unless you delve into HTML5 Canvas or SVG path animation and some scripting.