Non-parametric test for unequal samples with subsequent post-hoc analysis? Is it okay to perform a Kruskal-Wallis on four unequal samples? Further to this, is there a subsequent pair-wise post-hoc suitable for two unequal sample sizes?
 A: According to the formula for the Kruskal-Wallis test statistic, each group can have a different number of observations, so "yes".
Whether this is the best test or not I'm not sure - if you're still in doubt you'd need to post more details.  But perhaps this is all you needed to know.  Good luck!
A: Yes and Yes. Kruskal-Wallis analysis does not require equal sample size. In latest SPSS versions (from 18, if I remember correctly) there is a new nonparametrics procedure that performs pairwise comparisons with sig. adjustment, as well as step-down post-hoc method. Alternative would be to use very nice macro by Marta Garcia-Granero http://gjyp.nl/marta/
A: In case of heavy unbalanced groups the Kruskal-Wallis-test may be far off and you should not use it. There is a recent paper published on arXiv by Brunner et al. 2018 "Ranks and Pseudo-Ranks - Paradoxical Results of Rank Tests" in which the authors show that under certain conditions Kruskal-Wallis-test for more than two groups with unequal sample sizes may lead to intransitive decisions and false rejection of the null hypothesis. 

This  means  that  for  the  same  set  of  distributions $F_1 ,..., F_d$ and unequal sample sizes the p-value of the test may be arbitrary small if $N$ is large enough.

A possible solution is provided by the concept of pseudo-ranks. Apparently, the authors have implemented solutions (which I have not tested yet) for this in R and SAS. See this reference for more information: 
Brunner, E., Bathke, A.C., and Konietschke, F. (2018). Rank- and Pseudo-Rank Procedures for Independent Observations in Factorial Designs - Using R and SAS. Springer Series in Statistics, Springer, Heidelberg. 
A: Yes, Mann-Whitney tests are the normal post-hoc test to use for a Kruskal-Wallis test.
