Once a bank account is opened, it remains the property of the account holder, as long as it is active. The account holder can continue to deposit and/or withdraw money as before, as long as the bank's requirements for proper identity documents are satisfied. Visa status should not affect the account ownership or access; The bank will be unaware of the visa's expiration. The account holder will have access to the account, whether from the USA or from abroad. [EDIT ADDED: Procedures for sending instructions to the bank may become more elaborate, requiring authentication by the U.S. Consulate.]
Inactive accounts go through several stages. Initially, after some period, the bank will treat any funds according to its internal procedures and account rules, and may invest the money as if it were a long-term investment; The ownership remains with the account holder, but there may be procedural or practical delays in getting access to the funds. After a longer period, under state or federal regulations or laws, the state may take control of the funds, although they still will remain the property of the account holder.
Keep your account active, by depositing or withdrawing funds every month or two.
