While extensive work has been done bounding rates of convergence of highly symmetric and/or reversible Markov chains, less is known about the convergence behavior of arbitrary non-reversible chains. We give detailed descriptions of the long-term behavior of some simple families of non-reversible chains; these families have many deterministic transitions and underlying graphs "close" to a one-way cycle. By computing stationary distributions and approximating higher-order transition probabilities, we obtain local limit theorems for the distributions of these chains prior to stationarity. In all cases considered, the time to arrive at a fixed distance from stationarity is asymptotically $O(n^3 / m(n))$, where n is the total number of states and m(n) is the number of states with more than one possible successor.