I have called my state a "one shot" state, because a bank levy affects funds in the custody and control of the garnishee at the date and time that the levy is imposed. Those funds are frozen and potentially the property of the judgment creditor, subject only to claims of exemption and superior creditors' claims; "later arriving funds" are not affected. I'm aware of other states permitting a recovery order (call it writ, garnishment or whatever) to operate for a longer period of time, but I know nothing over 30 days.
The quoted Illinois statute would accurately descibe my state's approach to levies (property held at the time of service). In my state, the answer to the OP's question - can they take anything that goes in after that?- would be "no, not without a new levy" (but, it there's a new levy, then yes, everything that's non-exempt). My question - the OP's question - is does the Illinois statute mean more? The response posted by racer72 is "yes". If so, what's the authority? I'm not arguing; I simply want to know from those who subscribe to a different statutory construction.
I'm genuinely interested because, if the responses from racer72 and justalayman imply that a bank garnishment, once imposed, operates until the judgment is satisfied (and that's what was said), hypothetically, a non-wage garnishment in Illinois outlives a judgment. For example, if a judgment has been renewed as often as permitted by law, is unsatisfied notwithstanding and a bank garnishment is imposed in the final hours of the life of the judgment, those responses say it continues to affect the bank account (or did I miss a "but if" or "except when"?). That's a very interesting and unique proposition, and I'd like to better understand that statutory right.