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](https://doi.org/10.1016/j.chaos.2021.111413) or [Cánovas - On $\omega$-limit sets of non-autonomous discrete systems](https://doi.org/10.1080/10236190500424274), or [Cánovas - Recent results on non-autonomous discrete systems](https://doi.org/10.1007/BF03322551) (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