The follow path constraint place the constrained object on the selected curve considering a parameter relative to the curve’s length. this will cause the constrained objects to slide when the curve deforms.
To understand what I mean just constrain an object to a curve with multiple segments, set a very small offset (something like -2 or 0.2 if using the fixed position parameter), start pulling the vertices on the opposite end of the curve and you will see the object sliding.
It would be nice to have the option to attach the object on a U parameter, this way the position on curve will stay consistent regardless of the length of the curve.
Cheers
Maurizio