  We consider a one-dimensional continuous time random walk with transition
rates depending on an underlying autonomous simple symmetric exclusion process
starting out of equilibrium. This model represents an example of a random walk
in a slowly non-uniform mixing dynamic random environment. Under a proper
space-time rescaling in which the exclusion is speeded up compared to the
random walk, we prove a hydrodynamic limit theorem for the exclusion as seen by
this walk and we derive an ODE describing the macroscopic evolution of the
walk. The main difficulty is the proof of a replacement lemma for the exclusion
as seen from the walk without explicit knowledge of its invariant measures. We
further discuss how to obtain similar results for several variants of this
model.
