AddMovementInput requires a Pawn to work. The Pawn requires also a MovemenetComponent so it could also fail on custom Pawns where you forgot to add that component.
What I would do is to change the Actors WorldLocation (Set Actor World Location I think), this will change the location of the actor. You can also use a timeline and then make a linear interpolation to make the movement smoothly. Another way is to use the InterpTo function for the vector, there you can specify the current vector (the current actors location), the target vector (the one you will change), the current delta seconds and the speed at which the interpolation will operate. This will give a very nice and smooth movement for those actors.
Make sure you handle collisions which could block your movement.
You should also create your own Blueprint class and not a plain StaticMeshActor, those are meant to be static and not be moved.
EDIT: I’ll resolve the issue for now, if you need any further assistance please reopen the question.