You need to retrieve the Curve2D from the path and get the points (not the baked ones), then tween/move from point to point.
The moving node will need to keep track of where is in the curve (point index).
Another option is to add a PathFollow to move along the path.
I personally don't like the way PathFollow works because it needs to be child of the path and parent of the node it needs to move, like:
I would prefer some decoupling from the path and target node, but is the way it works, at most (without extra coding) you can do:
| |-RemoteTRansform2D (connected to Node2D_to_move)
Node2Dtomove may need
export(NodePath) var path_to_PathFollow
to be able to move it (just tween the offset/unit_offset).