ABSTRACT: We consider a random walk jumping on a dynamic graph; that is, a graph that changes at the same time as the walker moves. Previous works considered the case where the graph changes via dynamical percolation, in which the edges of the graph switch between two states, open and closed, and the walker is only allowed to cross open edges. In dynamical percolation, edges change their state independently of one another. In this work, we consider a graph dynamics with unbounded dependences: Glauber dynamics on the random cluster model. We derive tight bounds on the mixing time when the density of open edges is small enough. For the proof, we construct a non-Markovian coupling using a multiscale analysis of the environment. This is based on joint work with Andrea Lelli.