You could use Shapekeys
You could use hooks (Ctrl+H) attached to the control points and keyframe animate the movement of the hooks. This could just be manual movement or animate the hook following a curve
You could use the AnimAll addon http://wiki.blender.org/index.php/Extensions:2.6/Py/Scripts/Animation/AnimAll
Hooks definitely what I wanted here. Just add a hook to the control points you want to animate, move the hooks and and key frame them, and use the curve as a modifier or whatever. You saved me probably half a day of tutorials and barking up the wrong tree.