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 [dSOV21]; there it was shown that the herds process has a phase transition with respect to the infectivity parameter λ, depending on the parameter v that governs the edge dynamics. Improving on a result of [dSOV21], we prove that the critical value of λ is strictly decreasing with 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.