Pre-compose the scrolling images and mask the result in the parent comp or use track mattes. I strongly suggest you actually read the online help on some basics rather than directly delving into a downloaded template project and modifying it when you really don't know some basics, no offense.
Of course you may need more than one such matte if you have multiple layers. Similarly, you may naturally need to change your layout to accommodate your scrolling strips in the pre-comp. It simply seems you have locked yourself into that typical beginner thinking of trying to do it all in one comp with as few layers as possible. That's just not going to work. I would really repeat my point: Start by reading the online help on this stuff and then develop a strategy how you want to layout your animation. Many things in AE cannot be understood intuitively by trial & error and clearly you have navigated yourself into a corner here.
Pre-compose your image layers to a towel-strip sized comp, put it into your phone screen comp. Create your screen shape on top, select it as a track matte in the image-based pre-comp layer. Animate the position of the strip. Done. Please do read the online help on the basics.