The question really becomes, what is being evaluated in international sporting competition? Is it the ability of a certain ethnic group or a particular country's citizens? Or is it a country's sporting culture and the expertise/resources to develop young athletes to play a certain sport? Different governing bodies of particular sports place differing emphasis on these various factors.
Culture clearly influences people in these matters. People from more ethnically/culturally homogeneous countries tend to identify along those lines and have no issues welcoming members of their diaspora who've learned the game abroad. People from more ethnically/culturally diverse countries are less influenced by such matters. Neither side is "wrong", but the emphasis on ethnicity/culture does have an issue in my eyes.
If a country with no heritage in a particular sport either "imports" players from, or sends a large amount of young citizens to, a country that's world renown for developing players in that sport, then becomes a world power without any domestic ability to develop those players, is there much to be proud of? Conversely, if that country works hard on their own to develop the methods to create those players, wouldn't it be that much more satisfying and prove that much more about that country if they could find success?
Of course, most sports were historically created in one country and at some point, for the game to grow, require the influence of that country on others to foment interest/enthusiasm for the sport. Success is the best way to breed interest/enthusiasm in any given country for a sport, so if at some point, using players that were not developed domestically leads to the growth of the game in that country, the ends might justify the means.
It's not an entirely clear issue but, given the passage of time since many of the worlds popular sports were created, I think the emphasis really needs to move towards development over ethnicity/culture. I don't expect there will ever be an entirely "pure" system in this heavily globalized world but it raises the value of the competition in any given sport to continue working towards such a system. At the very least, I think it makes perfect sense for an athlete to spend some amount of time in the country they wish to represent before they're eligible to do so.