Let $(T,d)$ be an unbounded and locally finite metric space that contains a distinguished element o. Let $L^{\infty }\left(T\right)$ denote the discrete Banach space and $L^0\left(T\right)$ its subspace of functions vanishing at infinity. First, we establish that the recurrence of a composition operator on the $L^0\left(T\right)$ space is equivalent to its rigidity. We then provide an equivalent characterization for the uniformly rigid and $\mathcal{F}$-recurrence of $C_{\phi }$. For the $L^{\infty }\left(T\right)$ space, the recurrence of the operator is equivalent to the periodicity of the function $\phi$. Second, we explore the concept of disjoint topological recurrence associated with multiple composition operators.