There is a great tool for doing this. It is called Path/Sweep in the Animation Module.
I recommend looking in Refrence Manuals to get the workflow. However, I will give you the basic idea.
Essentially the command takes your shape (cross-section) and extrudes it along a path. Then animate the the cross-section by rotating it. When you play back the frames of the rotation, it literally deforms the shape of the extrusion. You'll have to do it to get the full impact of the idea, but you can make some really nice shapes with this too.
Here is a quick test:
Place a circle and rotate it so it is visible in top view (rotate 90 in X). This will be you path.
Place another circle. Leave it at 0,0,0, but scale it so that it is longer in Y. Scale the entire shape down so it is small enough to be a cross-section for an extrusion along the path. Else scale the path larger.
Select Path/Sweep in the Animation Module. You will be prompted for cross section then path. Then the amount of frames the effect occurs. Use Default 100 frame for now.
It doesn't look like much more than a extrusion along a path but....
Select the cross section in the front view. Save a rotation key for Z. Then rotate the shape several turns and save another key.
Now if you play the time line, you should see the cross section rotate, but you will also see the sweep change shape relative to chagnes in cross section.
Note: You can save lots of ratation keys, or even scale or do shape keys to make the cross section change. In your case the path will probably not be closed. Remeber to copy the sweep or freeze it after you get your desired results.
Hope this helps,
Urp