Move spline attached to gameobject.

[Juwdah]

If you set the Curvy Generator also as a child of the crane, your will get the behavior you expect.

The reason behind the behavior you are facing is that Curvy Generator uses the local positions of the input splines. So when you move the crane, the world position of the control points changes, but their local positions stay the same, thus the generated mesh stays the same.
When the input splines and the generator have the same parent, or even better, the spline is a child of the input spline module, like in the generator's "Shape Extrusion" template, the issue does not appear because moving/rotating the generator will also move/rotate the generated mesh.

Does my answer indeed fix your issue?

Yes! Thank you!

Is there a way to make this more performant? When animating this in unity there's lagg when playing this in the editor. And this is just two points being animated. The lagg stops when turning the generator off.

In play mode it laggs behind a bit. When the animation resets it resets one frame later.