# What methods can be used to compare the consistency of the distributions sampled from independent Markov Chains?

I have a set of independent random walkers on a graph that produce a distribution of the nodes in that graph. I would like to test whether the random walkers have produced a distribution which is consistent.

I thought about separating the results from the random walker into various sets, eg 4 separate groups, and using the distribution that these 4 iid groups make from the graph, use Gelman-Rubin diagnostics link to def.to test for convergence of those distributions.

I cannot use Gelman and Rubin's potential scale reduction factor directly on the independent walkers, because they are not let to run till they reach ergodic steady states. I suppose that the reasoning behind using Gelman and Rubin still holds when comparing within $W$, and between variance $R$ of distributions from groups of these independent walkers because even a markov chain run till it is supposed to reached ergodicity is a distribution.

Is my approach valid and why? are there any references for it?