They absolutely do, it is called "stand-in processing". I saw this while working in ATM at a major bank. The terminal operator (e.g. in our case, the ATM authorization system) can stand-in for the payment network when required. There are per-card number transaction limits that are well-defined in their contracts, and fraud liability can shift during this period of time. The payment network can also stand in for the issuer. In either case, once the network is restored all the authorization advices are forwarded.