2D parallax background

Hello. I´m making a 2d game where you enter rooms with a static camera, and I´m trying to implement fog through the parallax background technique (or in this case, foreground). I have an image that represents the fog and what I need is it to move over time so I thought that it could be done with a material that shifts the pixels to the right. I would like to know if it is possible and how I can do that.

Thanks in advance.