Hi!
I would have liked to see the Hierachy of the control a bit better in the video, so I could determine what exacly is wrong with it.
But from what I see in the video, you have a double hierarchy on your translates on the curve.
You can clearly see that there is conflicting influence on there because the indicators next to your values in the channelbox are green. You would want those indicators to be either red (keyframe), dark blue (set driven key), light blue (constraint) or yellow (directly connected), but not green (two or more of the others at the same time).
This comes most likely from the fact, that you have constrained and keyed it at the same time and maya tries to somehow blend between the two influences, but in my experiences, this almost never works out as intended.
Maya doesn't consider a constraint to be gone when all the weight values are set to 0 but it still considers the object to be constrained, but just to the point where it was initially constrained to in its local space. The constraint is active and has influence on the object at all times.
Normally you still can move your constrained object around in the Scene. But your Preferences->Animation-> Blend setting is probably on "Never blend with existing influences" or "Blend with everything except Constraints" in which case you cant use your move tool on constrained objects. But that doesn't really matter because either the animation curve node is overwritten by the constraint, so even if you could animate it, it would just snap back as soon as you move around in your timeline or move the constraint.
Or the animation curve node overrides the constraint, and when you move the influence object it is constrained to, nothing happens.
If you want to have an object constrained at some points in the timeline but animated in others,I would recommend you to introduce what is called a "free moving control" or "animation control", which is a second object, often a nurbs curve, which stands at the same place in 3D space and in the hierachy as your control curve and is also a constraint parent of your control, so when you want you object to be "not constrained", you simply switch all other weights on the constraint to 0 and the one on the "free moving control" to 1. And then animate the translates/rotates of your curve by using the "free moving control". That way you don't have hierachical conflicts and it will be a lot more predictable.
Hope this helps.
