A Markov channel is a discrete information channel that includes as special cases the finite state channels and finite state codes of information theory. Kieffer and Rahe proved that one-sided and two-sided Markov channels have the following property: If the input source to a Markov channel is asymptotically mean stationary (AMS), then so is the resulting input-output process and hence the ergodic theorem and the Shannon-McMillan-Breiman theorem hold for the input-output process. Kieffer and Rahe also provided a sufficient condition for any AMS ergodic source to yield an AMS ergodic input-output process. New conditions for a Markov channel to have this ergodicity property are presented and discussed here. Several relations are developed among various classes of channels, including weakly ergodic, indecomposable, and strongly mixing channels. Some connections between Markov channels and the theory of nonhomogeneous Markov chains are also discussed.