I assume that by "finitely many steps" you refer to the function composition operation. If so, the general direction at which you can look is probably the theory of attractive fixed points of discrete non-autonomous dynamical systems. For example look at this or this, or this (just to indicate a few among many works).

I assume that by "finitely many steps" you refer to the function composition operation. If so, the general direction at which you can look is probably the theory of attractive fixed points of discrete non-autonomous dynamical systems. For example look at Navascués - New equilibria of non-autonomous discrete dynamical systems or Cánovas - On $\omega$-limit sets of non-autonomous discrete systems, or Cánovas - Recent results on non-autonomous discrete systems (just to indicate a few among many works).

Not sure whether this is really helpful, but the general direction at which you can look is the theory of attractive fixed points of discrete non-autonomous dynamical systems. For example look at this or this, or this (just to indicate a few among many works).

I assume that by "finitely many steps" you refer to the function composition operation. If so, the general direction at which you can look is probably the theory of attractive fixed points of discrete non-autonomous dynamical systems. For example look at this or this, or this (just to indicate a few among many works).