When conducting a Kruskall Wallis test, it is possible to follow up the procedure with a post hoc test, looking for homogeneous subsets. My question is which statistic is used to perform this procedure (e.g., is it Chi-Square, z, or the original H)?
You are looking for Dunn's test, or the less well known, but more powerful Conover-Iman test. Both post hoc tests (a) perform pairwise comparisons using the same rankings used in the Kruskal-Wallis test (as opposed to just performing a bog-standard rank sum test for each pairwise comparison), and (b) use a pooled variance estimate implied by the Kruskal-Wallis test's null hypothesis. Dunn's test is based on an asymptotic z distribution, while the Conover-Iman test is based on an asymptotic t distribution.
I am unsure what you mean by "stepwise" or "step down", but implementations of both tests for R (dunn.test and conover.test) and for Stata (dunntest and conovertest) include step up and step down family-wise error rate and false discovery rate control for these tests.
Conover, W. J. and Iman, R. L. (1979). On multiple-comparisons procedures. Technical Report LA-7677-MS, Los Alamos Scientific Laboratory.
Dunn, O. J. (1964). Multiple comparisons using rank sums. Technometrics, 6(3):241–252.