Dragging breaks history?

No, I’m not at all familiar with.
Moreover I hardly grasp what you are talking about. Try to use the correct (or at least comprehensible) terminology – eg what is a “centerline” curve?
So please do post a sample model.

Maybe you’ll find a solution in Shifting pipe 001.3dm (88.0 KB)
As I explained in How to animate a curve point the technique is called “Morphing”.

Luc