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][1] or [this][2], or [this][3] (just to indicate a few among many works). 


  [1]: https://www.sciencedirect.com/science/article/pii/S0960077921007670?casa_token=eftiPyUWqFQAAAAA:4LEJ89Pd67zemlMavWkg8EaAY7zyxFbkE8ArNDyjAr_z_jlKQxn2C1kuN2ta7BBS05q6bb2B
  [2]: https://www.tandfonline.com/doi/full/10.1080/10236190500424274
  [3]: https://link.springer.com/article/10.1007/BF03322551