Two-sample Kolmogorov-Smirnov discrete distributions Let's say that two Census takers interview a number of couples and ask for the number of children they have. I want to compare the two resulting samples to find out if they could have come from the same population (let's say, the same country) or if they are more likely to have been drawn from two different populations.
I have been searching for quite some time and still could not find a definitive answer. Most sources state that the two-sample Kolmogorov-Smirnov test could be used if these were continuous distributions, which is not the case.
It seems that the two-sample chi-squared test, or the Wilcoxon-Mann-Whitney rank sum test could be used for discrete distributions, but these seem to suffer in the presence of many ties, which would be the case (most couples will have 0-3 kids).
Some have suggested the Wald statistic, which seems to be applicable to Poisson distributions. However, I don't assume that my problem follows any particular distribution.
In summary, is there a most appropriate test to find out if these two discrete/count samples come from the same population?

You can certainly use a two-sample Kolmogorov-Smirnov test with a discrete distribution. But it is possible that the typical asymptotic approximation used in many implementations will potentially require a much larger sample than would be needed with a continuous distribution. Regardless, the simple solution to that problem is to find an implementation that is randomization based or uses some other approximation for the p-value.
If you are using R, I believe the packages twosamples, KSgeneral, and dgof all appear to have KS test implementations that should work. I recall several others, but I can't find them at the moment.

ETA: I don't recommend immediately jumping to Chi-sq -- though it will also work in these situations. This is because Chi-sq doesn't have any understanding of ordering in the data. If one observation moves from 0 kids to 5 kids between samples, chi-sq will only see 1 observation moving, whereas KS will see 1 observation making a big jump. Having said that, non-permutation KS tests are notoriously underpowered, so Chi-sq may often be a good option -- even though it gives up a lot of information, it doesn't waste anything it actually uses.
