I have a quasi-experiment where I'm comparing one experimental group (A) to many control groups (B1, B2, B3, B4, B5). Since this is a quazi-experiment, I do not know for sure that my control groups all come from the same distribution. I want to know whether some values for the experimental group are significantly different from the control groups. For some values, I hypothesize there will be a difference, and for other that there will not be.
My current approach is to use an omnibus test across all groups (e.g. ANOVA, Kruskal-Wallis) and then post hoc pairwise tests (e.g. Tukey's HSD, Dunn test). This does allow me to detect differences between the groups, but it makes interpretation more difficult. For example, what if A > B1..B3, but there are no differences between A and B4 or B5? Or what is B1 > B2?
Is there a better test for investigating my hypotheses? Or would a modeling approach be more appropriate (e.g. HLM)?