"[Card's] latest paper shows that cities with high rates of unskilled immigration have reported no offsetting shrinkage in the number of native-born laborers. "
Card is forgetting to look at the opportunity cost to American citizens. Immigrants don't move to American cities with lousy economies, they follow the money to cities with growing economies. Those cities shouldn't have "non-shrinking" numbers of Americans working in them, they should have growing numbers of native-born workers. But with mass immigration, the immigrants crowd out American citizens who would otherwise move there.
Then, when the economy turns down, the American citizens flee. That's exactly what happened in California: for generations, American citizens moved to California for the good life, but the flood of illegal immigrants made that too expensive. Then, when the economy turned down in the mid-1990s, the natives fled to other states.
Update: The David Card study is Is the New Immigration Really So Bad? [PDF]