By means of a new technique, we develop further a discrete subordination approach to the functional calculus of power bounded and Ritt operators initiated by N. Dungey in [19]. This allows us to show, in particular, that (infinite) convex combinations of powers of Ritt operators are Ritt. Moreover, we provide a unified framework for several main results on discrete subordination from [19] and answer a question left open in [19]. The paper can be considered as a complement to [26] for the discrete setting.