Maybe I'm missing something but this seems like a pretty perfect application for the bootstrap - a remarkably intuitive but powerful framework. Without loss of generality, imagine that you have two populations, A and B, and that you want to test some hypothesis about a statistic of A being different from a statistic of B (mean, in this case). Using the simplest form of the bootstrap you would do the following:

1. Pool and randomly label the data from A and B 2. Sample with replacement and form two partitions of the same cardinality as the original A and B groups 3. Compute the differences in mean 4. Rinse and repeat millions of times to form a distribution of mean differences 5. Check if the observed difference in means (from the true A/B labels) is statistically significant relative to the distribution found in (4)

This has some problems with fat tailed distributions but tends to work great otherwise. It's so simple that it avoids a host of pitfalls that can arise with other resampling schemes (what's being proposed is a type of resampling), and I love that it makes basically zero assumptions on the underlying data.