Moving a single vertex along a path or in a curve

If you have a path (mask or shape), you can use the Window/Create Shapes from Paths/Points follow Nulls script to add a null to every point on the path. You can then delete the nulls you don't need and animate the position of a null or the remaining nulls that are left. 

If you want to draw a path that one of the nulls will follow, create a new shape layer, draw the path with the pen tool, reveal the new path in the Timeline, set a keyframe for the path, then cut the keyframe, select the position property of the null, add a keyframe, then paste. Setting a keyframe instead of just copying the path guarantees that you have the path at that specific time copied, then setting a new position keyframe guarantees that the new motion path will start where you want it to start.

If that is not what you want to do, let us know. You can always edit motion or vector paths using the pen tool and the Alt/Option and Ctrl/Cmnd modifier keys right in the Composition Panel.