I've seen transethnicity most in people of mixed heritage, where the visually prevailing racial traits clash with the individual's internal identity and/or cultural upbringing. Make no mistake: this can be just as devastating and potentially fatal as any other disconnect. It's not okay to make fun of. The fact that some people practice cultural misappropriation in ways that look vaguely similar does not make this problem any less real or harmful, just as people's occasional mocking crossovers do not make the transgender thing any less so.
It shows up in some instances more than others, partly for cultural reasons and partly for genetic ones. Some traits are just more persistent than others. One of the more memorable examples I remember is from an Australian documentary about the indigenous people there, in which a man described himself as, "I'm a blackfella in a whitefella's body." And he was; you could just barely see the aboriginal ancestry he had, but there was more of it in his family tree than really showed. For some reason that's fairly common; the Australian features seem to fade quickly when mixed with other gene pools, whereas some others like African are more persistent. It's popped up in other references. That bothers some of the mixed-race descendants. Then there are cultural factors like the African-American tussle over whether someone is "black enough." There are several Native American prophecies about people being reincarnated from tribal to other cultures, sometimes European but other times global in scope, with their original tribal affinity intact. That hasn't seemed to lessen the hate-on much at all. The issue of "passing" appears in most if not all mixed-race contexts. Many hints or bolder statements about this sort of thing appear in various branches of ethnic literature, mostly by mixed-race writers. So people can be of any appearance, and feel that it doesn't fit who they really are.
From what I've seen, tolerance of transethnicity approaches zero, much like transgender. People will defend those lines literally to the death. Which is stupid, because race doesn't even have as much biological backing as sex does; most of it is essentially cosmetic and doesn't correlate with major population division factors. Go figure. But in any case: if someone feels that their soul and their body don't match up, that tends to be damaging. Making it worse by picking on them is not a decent thing to do, nor does it particularly help settle the troubled waters of race relations. It's just a nasty mess that hardly anyone has put any effort into figuring out how to relieve.
[EDIT 7/22/12] Happily my audience seems to be saner on this topic than other folks discussing it. (You are so awesome.) Here is a post that has spun off into another blog:
"Once Again" by marina_bonomi