Constrain vertex positions to moving objects? I have a schematic of a mechanism on which I would like to superpose an animated 2-D line drawing (a simplified representation of the mechanism). 

I'm struggling with a few things, including how to draw the 2-D lines without faces. For now, I've decided to draw a single polygon (which I can render as a wireframe with Freestyle or with the Wireframe modifier).

But to make this work, I need the vertices of the polygon to move with the joints of the underlying mechanism. The edges between the vertices would correspond to the parts connected through joints, and they would rotate and translate as the vertices move around. 

All I know at this point is that I can move the (here polygonal) mesh origin to a vertex and constrain the mesh position to another object. But this leaves me with all the other vertices that I'd like to constrain in a similar manner. If I simply move the mesh origin to another vertex to constrain that to another object, the first constraint changes and I'm back to step 1. 

Thanks in advance for whatever help you can give!
You can constrain each individual vertices of your polygon to the joints of your mechanism instead of moving the mesh origin. No need to move the mesh origin.

You can do it using the Hook modifier. Create one Vertex group for each vertices of your polygon, then, using the Hook modifier, constrain each of those group to one of the joint of your mechanism, using whatever mesh that makes up the joint that you think would show the best result.

Alternatively, you can use some Empty, constrain them to the center of the joints, then Hook each Vertex group to one of the Empty.