If it's a reschedule before the flight is under airport control, or before the passenger has checked-in, it is generally down to the issuing airline or travel agent as far as I understand. That's when we get the line from the operating carrier 'oh, we can't touch this'.
After airport control or check-in, it is the operating carrier.
We see this time and time again with Thai Airways on award tickets... they have a schedule change and misconnect the passenger, but when the passenger calls thai, they refer us back to the award issuer (US airways or united etc). The passenger can be caught in limbo if they don't know how to handle it.