Memristor Cellular Nonlinear Networks (M-CNNs) represent a significant leap in computational technology compared to traditional Cellular Nonlinear Networks (CNNs), thanks to their multi-tasking and memcomputing capabilities. Recent studies have demonstrated various configurations of M-CNNs that utilize these capabilities to perform image processing tasks. This paper employs the Dynamic Route Map circuit-theoretic analysis tool to investigate the dynamic features of M-CNNs and shed light on the underlying mechanisms responsible for their ability to handle multiple tasks. The findings from this theoretical study offer valuable insights for the development of more compact and highly efficient data processing M-CNNs that possess such versatile properties.