Only this is less common than the other.
Besides, for those anti-immigrant, whether "born here" or "newly arrived" doesn't matter much. If they are not of English, Scottish, etc ancestry, they are immigrants.
So saying that the e.g. the terrorists in Germany were German because they were born in the country is a little beside the point when they are clearly from middle-eastern origin for example.
Well, now here we have a problem, because immigration policy can only apply to people who are legally considered immigrants. The word for expelling large numbers of people who were born in the country but not of the dominant ethnicity is "ethnic cleansing".
In London, if population trends are any indication, this sentence applies just as well to English people...
They might not want them expelled, but simply counted along with new immigrants in estimations of the cumulative amount of people of foreign ethnicities that resulted by immigration (and thus in decisions whether immigration has been "too much" or not).