Recent work has considered a class of cellular neural networks (CNNs) where each cell contains an ideal capacitor and an ideal flux-controlled memristor. One main feature is that during the analog computation the memristor is assumed to be a dynamic element, hence each cell is second-order with state variables given by the capacitor voltage and the memristor flux. Such CNNs, named dynamic memristor (DM)-CNNs, were proved to be convergent when a symmetry condition for the cell interconnections is satisfied. The goal of this paper is to investigate convergence and multistability of DM-CNNs in the general case of nonsymmetric interconnections. The main result is that convergence holds when there are (possibly) nonsymmetric, non-negative interconnections between cells and an irreducibility assumption is satisfied. This result appears to be similar to the classic convergence result for standard (S)-CNNs with positive cell-linking templates. Yet, due to the presence of DMs, a DM-CNN displays some basically different and peculiar dynamical properties with respect to S-CNNs. One key difference is that the DM-CNN processing is based on the time evolution of memristor fluxes instead of capacitor voltages as it happens for S-CNNs. Moreover, when a steady state is reached, all voltages and currents, and hence power consumption of a DM-CNN vanish. This notwithstanding the memristors are able to store in a nonvolatile way the result of the processing. Voltages, currents and power instead do not vanish when an S-CNN reaches a steady state.