> I don't know what to do if another change interrupts the first but it's a rare case and can probably be handled imperfectly.

Extract the position (p0) and velocity (v0) vectors at the moment of interruption, and derive a new function F(t) that meets the constraints {F(0)=p0, F'(0)=v0, F(1)=p1, F'(1)=0}.