It was a little trickier than I thought it would be, but this works:
Overview: Copy Eraser Shape, paste it into a new mask Shape, copy that into Position.
Details:
On the layer with Eraser, hit “UU” and any properties changed from normal will be revealed. The Eraser movement info is hiding in the Eraser 1> Shape parameter. So select “Shape” and Copy.
Now in the layer you want to follow the eraser path (I used a small solid), draw a 2 point mask (G gives the Pen Tool, 2 clicks for a 2 point mask) make it anywhere on that layer.
“M” reveals the mask, select the “Mask Shape” and Paste (pasting that Eraser Shape into the mask shape).
Select that mask Shape and Copy.
Now “P” reveals the position (still on that layer we want to trace the path), select the Position and Paste.
You should be done, This is assuming that your layer being erased and the layer following the path had the same position to start with. If both are centered in the comp, OK, or just make the new Moving layers psoition the same as the Erased layers position when you start. Also the position path will come in as a 2 second path with roving keyframes so grab either end position keyframes and drag it left or right as needed to make the animation the length of time you want.