We study the contact process on a dynamic random~$d$-regular graph with an edge-switching mechanism, as well as an interacting particle system that arises from the local description of this process, called the herds process. Both these processes were introduced in~\cite{da2021contact}; there it was shown that the herds process has a phase transition with respect to the infectivity parameter~$λ$, depending on the parameter~$\mathsf{v}$ that governs the edge dynamics. Improving on a result of~\cite{da2021contact}, we prove that the critical value of~$λ$ is strictly decreasing with~$\mathsf{v}$. We also prove that in the subcritical regime, the extinction time of the herds process started from a single individual has an exponential tail. Finally, we apply these results to study the subcritical regime of the contact process on the dynamic $d$-regular graph. We show that, starting from all vertices infected, the infection goes extinct in a time that is logarithmic in the number of vertices of the graph, with high probability.