Let me clarify this a bit, for on almost all of the VOA 24-hour FM transmitters which run VOA1 music programming, that local programming is not programming inserted locally at the transmitter location, but is in the satellite audio feed from the VOA studio complex in Washington. These VOA 24-hour FM transmitter sites typically have no local access to the audio stream, short of a physical access between the satellite receiver and the transmitter input, when the satellite receiver is at the transmitter location.
One exception that I am aware of is with the VOA FM on Sao Tome, where the audio for the FM transmitter on the mountain location is fed by an STL from the VOA station on the coast. The programming for that transmitter is usually the VOA1 music feed, but there were a few program changes, late afternoon and early evening, where the VOA1 is not run, but program segments from the Portuguese for Africa feed and French to Africa feed are used.
To the best of my knowledge, there is no VOA1 programming on any of the short wave transmitters.
The VOA affiliate operations, local broadcasters, have agreements to run VOA programming, as a minimum, they would have to run the programing per the agreement, but I don't think any of those agreements limited those local broadcastings to only those programming segments. Since the VOA satellite feeds are not encrypted, there are probably broadcasters using VOA programming and VOA HQ might not ever know about that program use.
David mentions tape programming, and many years back, even some of the VOA programming was distributed to the overseas VOA transmitter locations, on disk in the early days and later, magnetic tapes, programming which wasn't of a time-sensitive nature. This was because the audio feeds to the overseas transmitters were the shortwave transmitters in the United States, and propagation wasn't always great, so with these canned programs, the overseas transmitters would have higher quality audio available than what might be available via the short wave feeders. Once the VOA set up the satellite distribution network, there was no longer any need for the feeder transmitters at the stateside locations.