We develop a Markov process viewpoint for discrete circular distributions motivated by directional-statistics settings where angles are observed on a finite grid and evolve over time. On the $m$-point discrete circle, the cycle graph, we study diffusion-generated families, obtaining an explicit transition kernel, exact trigonometric moments, and convergence to uniformity. We present a simple approach to construct reversible nearest-neighbour chains with any prescribed strictly positive stationary pmf $π$, providing discrete analogues of Markov processes on the continuous circle. We construct processes whose stationary laws are the discrete von Mises and wrapped Cauchy distributions with closed-form normalizers and exact moments.