To start or change the movement, we need a force. This force, according to Newton's second law, is variation of quantity of motion from the object per unit of time.Therefore, if the variation does not exist, the amount of movement of objects is conserved.This conclusion allows enunciate the Newton's first law as follows: The amount of movement of an object is preserved if the resultant of the forces that act on it is null. So, Does the movement of things in the universe will cease one day?
Making predictions like these are impossible to be honest.
Furthermore it strongly depends on what you mean by movement ceasing. Assuming a non-collapsing Universe and the validity of QM the answer would be No. The uncertainty principle alone would be enough to justify this.