Suppose we have a circular arrangement (periodic boundaries) with $M$ sites and we want to distribute $N$ particles over these sites such that there are occupation numbers $n_m$ that respect $\sum_m n_m = N$. Now in each step every single particle stays with probability $p$ and a jumps with probability $q$ to the left or to the right (only to the neighbors). Therefore we have $p+2q=1$.

Now given the sets of occupation numbers $\{n_m\}$ and $\{n'_m\}$ of two consecutive steps, what is the transition probability between these two sets of numbers?

The challenge in this question arises because the "macro" combinatorics has to be worked out starting from the "micro" combinatorics for each particle, such that one has to consider several micro-configurations that give rise to the same macro-configuration though each of them has a different probability.

Maybe there is a nice reformulation of this combinatorics problem or an approach with stronger theoretical input, since basic counting has not gotten me past the micro <-> macro issue.