All social science research faces questions about the external validity of the results. Much social psychology research is done on students and so the natural question is whether those findings generalize to non-student populations. Even representative surveys of the population face questions about validity due to the assumptions which go into what representative means. Since all measurement is imperfect, one of the main ways to determine the robustness of a finding is to examine many measurements and look for overall patterns. FiveThirtyEight.com did this during the 2008 presidential election and became a national sensation.
The central finding of Moral Foundations theory to date is the split between what liberals and conservatives report caring about. Specifically, Liberals care more exclusively about issues concerning harm and fairness, while conservatives also care about issues surrounding obeying rightful authority, being loyal to one's ingroup, and avoiding "unnatural" violations of one's purity.
How can we tell if this finding is robust? All web servers keep track of referring traffic and so we can analyze the data we collect at yourmorals.org by the source of the traffic. If the pattern holds among people who read the New York Times, people who come from conservative blogs (a minority, but there are some), people who read the Houston Chronicle, people who find the site by typing 'morality quiz' into a search engine, and people who read Libertarian magazines....then it is likely that the pattern is somewhat robust. Of course, these patterns are all among internet samples, so it would be fair to say that if this pattern of liberal-conservative differences holds among all these groups, then it is fairly robust amongst the type of people who use the internet to read about news or politics.
Below are graphs across many of these groups. You'll see the same pattern where as you move from liberal to conservative, the exclusivity of concern about issues of harm and fairness gets less and less.