There's a zillion ways the Chechneyan bombers were being financed by unsuspecting Bostonians.... and other Americans. I'm sure they know the ropes.....probaly bought fake ID's to establish multiple identites.
Claiming "asylum from Russian persecution" gets tham a wealth of tax-paid freebies---no questions asked----a policy dating back to WW 11.
There's housing subsidies, instant welfare, UI, EITC, SSI, education payments, free medical care.....and so on.
The possible issues you raise would be good to investigate and publish, especially as part of the proposed “comprehensive” immigration reform.