About twelve years ago I worked for a company which processed debit card payments. The money for the transactions from little atm card machines would flow into our company account at a small local back via ACH, and the next day we’d provide a balanced ACH file to the bank with lots of small transfers to send out the reimbursements (less the per-transaction fees) to the clients.
In making sure the format for our ACH file was correct (we actually delivered them by hand on floppy disk, at first) I had occasion to visit the bank and watch the process. Remember, this is a small, local bank. (Eventually, we trained the bank to call in to our systems via modem and retrieve the prepared ACH file every morning. We paid 5 or 10 cents per ACH transfer; hundreds per day and maybe thousands on peak days and at month’s end when we paid commissions.)
In the back room they had an ordinary PC not on any LAN. It ran some software (for some reason I think it was called FedWire, or had something to do with that) which on a regular schedule, like twice a day, would call in via modem to the Federal Reserve to drop off and pick up ACH transactions. There was a dot matrix printer with two-part NCR paper which printed out each transaction, and the second copy was something that had to be saved forever, for auditing.
Now, if there was a wire transfer going out, the operator would set it up on this computer as a transfer of funds to debit from the bank’s Fed account and send via wire to a wire-receive-number account (not ACH). For a received wire, the Fed computers could call this PC anytime and make one happen - then the operator would have to notice that there was an incoming wire on the dot matrix printer, tear it off and go deal with the local bank’s books. Usually, in this bank, someone would come in and say they were expecting a wire and that’s when the teller, or head teller, would go check the printer, if it hadn’t been noticed.
IIRC, the funds would then be temporarily in the bank’s own Fed account.
It’s likely I got some of these details wrong through imperfect supposition, especially now, 12 years later and probably a light year away from how a major bank (BofA, Wells Fargo, Chase) operates - but I hope my skeletonized framework of understanding can carry its own weight.
I’d guess that a major bank has a full-time staff in a centralized wire-transfer department per region which takes the wire details and updates the bank’s own accounts via WAN, so even a remote branch doesn’t need something like the backroom computer I describe.