ALL the New Deal programs required that you go visit a caseworker if you moved ~ you can find out who has a forwarding order on file by cranking up a computer program that handles Move Update ~ see https://www.usps.com/business/move-update.htm
There is a sort of underground railroad for illegal immigrants. They start off in, let’s say NYC, then move westward via family contacts, collecting all the merry way.
They know all the ins and outs, and I don’t know of any incentives for any government entity to go and check - you know there’d be a hue and cry if there was any sort of bounty or bonus associated with catching someone.