I get WinSCP to right before the problem occurs, start recording, in one window, and start the transfers in the other. 1, 2, 3, 4... Something flashes by in Wireshark scrolling packets. 5 is stuck... Timeout.
Well that temporal clue is unusually handy, isn't it. Could be nothing. Let's go look.
!!! NOTE !!! This Wireshark also shows the computer talking with the server over TLS/443, ignore those they're irrelevant to this scenaro.