Move along a run-time generated path of varying length at a set speed.

I need to figure out how to move an object along a path a set distance or speed. For example, assume I have a curved path with a cube following that path. I want it to move 10 units along that path, or at least move toward the end of the path with a speed of 10, but only once each time I call a method.

I have looked at iTween, HOTween, DOTween and LeanTween. All of these seemed like they could have solved my problem, but they only allow you to set the time it will take to move across the entire path. This doesn't work for me because I am generating the path on the fly. I need some way to move along a path of any distance at the same speed as other paths.