I will assume you already have the animation of the screens completed. However, it's still difficult to say what you should do next as I am unsure as to whether you have animated using a camera that is moving in 3D space, or you are simply animating the screens' position properties to budge them in and out of the visible comp area.
If you are animating screen position, you should consider using a camera instead.
The reason I'm thinking you might be animating screen position is that, if you were using a camera, you would not have to worry about making your footage "stay in the screen" when the screen moves. The footage would share much the same position as the screen, and this wouldn't change when the camera moved.
However, if you have the animation just as you want it, place the footage beneath the screen in the layer stack, use the pen tool to draw a mask around the edges of the screen if the screen is not already transparent, set the mask to "subtract", position the footage how you want it, then parent the footage to the screen. When the screen moves, the footage will move with it.
Also remember to turn on motion blur for all your layers and for the comp.
Hope this helps.
Simon Bonner
