To play devils advocate: why shouldn’t we expel unproductive native-borns? They are as much a drag on the economy as unproductive non-native-borns.
Well I can only answer for myself, but my thought is that our ideals of citizenship must include acknowledging that some people are not capable of being productive - children, the infirm, elderly, etc. Some of these non-productive people will become productive, others used to be productive, all can add value to our society beyond the creation of goods and services. Stated differently, human relationships, including citizenship, are about more than economic productivity.
At the same time, if we are to continue with the idea of "government as provider", we cannot afford to increase disproportionately the number of people who are supported by those who are productive. Reasonable people can differ on the extent to which government is now expected to be a provider, but I think it's inarguable that the proportion of government spending on "entitlements" is larger, almost infinitely larger, than when mass immigration occurred in the 19th century and gave rise to the "nation of immigrants" concept. To consider immigration policy now without that consideration would be at best naive and more likely malfeasant, in my opinion.
I would further argue that entitlement spending itself should be seriously re-considered from the same perspective. Whether supporting resident alien, naturalized, or native-born, there is a limit to the degree of support John Q. Taxpayer can provide to those around him; $20 trillion in federal debt alone suggests to me that we are beyond that limit.