The random stirring process is a natural random walk on the set of permutations of the vertex set of a graph. The cyclic time random walk is a self interacting random walk on a graph. It is influenced by its past, in that it is constrained to repeat its past choices if it returns to a previously visited edge after a multiple of some period of time. The two models are fundamentally equivalent to each other as well as to a certain coalescence and fragmentation process.