Recent papers in the literature introduced a class of neural networks (NNs) with memristors, named dynamic-memristor (DM) NNs, such that the analog processing takes place in the charge-flux domain, instead of the typical current-voltage domain as it happens for Hopfield NNs and standard cellular NNs. One key advantage is that, when a steady state is reached, all currents, voltages, and power of a DM-NN drop off, whereas the memristors act as nonvolatile memories that store the processing result. Previous work in the literature addressed multistability of DM-NNs, i.e., convergence of solutions in the presence of multiple asymptotically stable equilibrium points (EPs). The goal of this paper is to study a basically different dynamical property of DM-NNs, namely, to thoroughly investigate the fundamental issue of global asymptotic stability (GAS) of the unique EP of a DM-NN in the general case of nonsymmetric neuron interconnections. A basic result on GAS of DM-NNs is established using Lyapunov method and the concept of Lyapunov diagonally stable matrices. On this basis, some relevant classes of nonsymmetric DM-NNs enjoying the property of GAS are highlighted.