This argument is part of the broader welfare objection.
The argument here is that immigrants (legal and illegal) as well as their citizen children, make disproportionate use of means-tested welfare benefits on account of their poverty. The claim isn’t specifically that immigrants immigrate to live off the welfare state, but rather that once they have immigrated, they discover that they can take advantage of various welfare benefits. Some people have also considered the second-order effects where the presence of a welfare state acts as a negative selection mechanism for immigrants, but this is not central to the argument.
For more country-specific and region-specific arguments, see: