On old gaming consoles (first gen) there were games made like FPS, but in which the background moves from right to left, and the scene could stop for some time and then go on. Also, the player could only move the crosshairs.
It's called being on "rails". You can see this in several games, NES Adventures of Bayou Billy stage 2 being one of the oldest I can remember. Revolution X being one of the later ones. In levels like this, you are on a "rail" being moved by the screen shooting at things. You don't see the rail and can only control your guns, but feel your movement to the right.