  We consider a dynamic random graph on $n$ vertices that is obtained by
starting from a random graph generated according to the configuration model
with a prescribed degree sequence and at each unit of time randomly rewiring a
fraction $\alpha_n$ of the edges. We are interested in the mixing time of a
random walk without backtracking on this dynamic random graph in the limit as
$n\to\infty$, when $\alpha_n$ is chosen such that $\lim_{n\to\infty} \alpha_n
(\log n)^2 = \beta \in [0,\infty]$. In [1] we found that, under mild regularity
conditions on the degree sequence, the mixing time is of order
$1/\sqrt{\alpha_n}$ when $\beta=\infty$. In the present paper we investigate
what happens when $\beta \in [0,\infty)$. It turns out that the mixing time is
of order $\log n$, with the scaled mixing time exhibiting a one-sided cutoff
when $\beta \in (0,\infty)$ and a two-sided cutoff when $\beta=0$. The
occurrence of a one-sided cutoff is a rare phenomenon. In our setting it comes
from a competition between the time scales of mixing on the static graph, as
identified by Ben-Hamou and Salez [4], and the regeneration time of first
stepping across a rewired edge.
