  We investigate the mixing properties of a finite Markov chain in random
environment defined as a mixture of a deterministic chain and a chain whose
state space has been permuted uniformly at random. This work is the counterpart
of a companion paper where we focused on a reversible model, which allowed for
a few simplifications in the proof. We consider here the general case. Under
mild assumptions on the base Markov chains, we prove that with high probability
the resulting chain exhibits the cutoff phenomenon at entropic time $\log n /
h$, $h$ being some constant related to the entropy of the chain, when the chain
is started from a typical state. However contrary to the reversible case
uniform cutoff at entropic time does not hold, as we provide an example where
the worst-case mixing time has at least polylogarithmic order. We also provide
a polylogarithmic upper bound on the worst-case mixing time, which in fact
plays a crucial role in deriving the main result for typical states.
Incidentally, our proof gives a clear picture of when to expect uniform cutoff
at entropic time: it appears as the consequence of a uniform transience
property for the covering Markov chain used throughout the proof, which lies on
an infinite state space and projects back onto the initial chain.
