For a finite commutative ring $R$ and a positive integer $k\geqslant 2$, we construct an iteration digraph $G(R, k)$ whose vertex set is $R$ and for which there is a directed edge from $a\in R$ to $b\in R$ if $b=a^k$. Let $R=R_1\oplus \ldots \oplus R_s$, where $s>1$ and $R_i$ is a finite commutative local ring for $i\in \{1, \ldots , s\}$. Let $N$ be a subset of $\{R_1, \dots , R_s\}$ (it is possible that $N$ is the empty set $\emptyset $). We define the fundamental constituents $G_N^*(R, k)$ of $G(R, k)$ induced by the vertices which are of the form $\{(a_1, \dots , a_s)\in R\colon a_i\in {\rm D}(R_i)$ if $R_i\in N$, otherwise $a_i\in {\rm U}(R_i), i=1,\ldots ,s\},$ where U$(R)$ denotes the unit group of $R$ and D$(R)$ denotes the zero-divisor set of $R$. We investigate the structure of $G_N^*(R, k)$ and state some conditions for the trees attached to cycle vertices in distinct fundamental constituents to be isomorphic.