  In a previous part of this work, we gave a new tableau formula for the
modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$, using a weight
on tableaux involving the queue inversion (quinv) statistic. In this paper we
establish a link between these combinatorial objects and a class of
multispecies totally asymmetric zero-range processes (mTAZRP) on a ring, with
site-dependent jump-rates. We construct a Markov chain on the space of tableaux
of a given shape, which projects to the mTAZRP, and whose stationary
distribution can be expressed in terms of quinv-weighted tableaux. We deduce
that the mTAZRP has a partition function given by the modified Macdonald
polynomial $\widetilde{H}_{\lambda}(X;1,t)$, and we obtain interesting symmetry
properties of the mTAZRP probabilities under permutation of the jump-rates
between the sites. We explore a number of interesting special cases of the
mTAZRP, and give explicit formulas for particle densities and correlations of
the process purely in terms of modified Macdonald polynomials.
