Let $G$ be a nonamenable transitive unimodular graph. In dynamical percolation, every edge in $G$ refreshes its status at rate $μ>0$, and following the refresh, each edge is open independently with probability $p$. The random walk traverses $G$ only along open edges, moving at rate $1$. In the critical regime $p=p_c$, we prove that the speed of the random walk is at most $O(\sqrt{μ\log(1/μ)})$, provided that $μ\le e^{-1}$. In the supercritical regime $p>p_c$, we prove that the speed on $G$ is of order 1 (uniformly in $μ)$, while in the subcritical regime $p<p_c$, the speed is of order $μ\wedge 1$.