There is no unmoving mover behind the movement. It is only movement. It is not correct to say that life is moving, but life is movement itself. Life and movement are not two different things. In other words, there is no thinker behind the thought. Thought itself is the thinker. If you remove the thought, there is no thinker to be found.
Walpola Rahula, What the Buddha Taught: Revised and Expanded Edition with Texts from Suttas and Dhammapada