The mixing time of a random walk, with or without backtracking, on a random graph generated according to the configuration model on n vertices, is known to be of order log n. In this paper, we investigate what happens when the random graph becomes dynamic, namely, at each unit of time a fraction αn of the edges is randomly rewired. Under mild conditions on the degree sequence, guaranteeing that the graph is locally tree-like, we show that for every ε ∈ (0, 1) the ε-mixing time of random walk without backtracking grows like 2 log(1/ε)/log(1/(1 − αn)) as n → ∞, provided that limn→∞ αn(log n)2 = ∞. The latter condition corresponds to a regime of fast enough graph dynamics. Our proof is based on a randomised stopping time argument, in combination with coupling techniques and combinatorial estimates. The stopping time of interest is the first time that the walk moves along an edge that was rewired before, which turns out to be close to a strong stationary time.