Situationist critiques of character (as discussed in Appiah's Experiments in Ethics appeal to psychological research to challenge the idea that there are general, stable character-based dispositions. A classic case is the phone booth study (Isen and Levin (1974)), in which individuals who found a dime in a phone booth nearly all helped a person who dropped a pile of papers, whereas those who went dimeless almost never helped. So, kindness, it would seem, is contingent on situation (and perhaps mood) rather than any kind of stable character trait as kindness. One can contest these sorts of critiques on various grounds--perhaps genuine character is the exception rather than the rule, for example, and so it shouldn't be surprising that the behavior of most people varies more radically.
So here's the good news I discovered for moral courage, in a paper from The Psychology of Courage: Modern Research on an Ancient Virtue. If we take mood to be the factor affecting helping behavior in the phone booth study, Niesta, Greitemeyer, Fischer, and Frey (2008) (cited in Osswald, Greitemeyer, Fischer, and Frey, in the volume linked above) found that mood does not affect morally courageous intervention. This may be because the perception of a greater wrong is more strongly motivating than just the perception of something, as it were, amiss. That seems right. (This sort of action, unlike helping behavior, involves incurring substantive risks to oneself in order to intervene in the unjust (etc.) treatment of another.) The article in which the study is cited doesn't say what proportion of subjects intervened in the courage situation, and that might be worth knowing, too, for other reasons.
Some less good (though perhaps not shocking) news from the article is that while people are typically quick at recognizing situations that would seem to call for a (morally courageous) intervention, "people feel less competent to intervene than in other prosocial incidents." That's important, insofar as confidence is essential to courageous action (and where calling the police isn't a viable option...) I wish I could read German, as the authors have several works in their native language, and conduct workshops that aim to equip people with reasonable strategies which at the same time minimize risk to self (in part by finding ways to involve others in the intervening action).