  We consider dynamical percolation on the complete graph $K_n$, where each
edge refreshes its state at rate $\mu \ll 1/n$, and is then declared open with
probability $p = \lambda/n$ where $\lambda > 1$. We study a random walk on this
dynamical environment which jumps at rate $1/n$ along every open edge. We show
that the mixing time of the full system exhibits cutoff at $\log n/\mu$. We do
this by showing that the random walk component mixes faster than the
environment process; along the way, we control the time it takes for the walk
to become isolated.
