Domesticated magpies will carefully watch over these flying livestock like smarter, louder, sheepdogs, driving away would be predators (along with the help of a much smaller number of domesticated eagles, falcons, etc). Without any human contact for up to months at a time these magpies will herd the livestock birds between regions, filling an analogous role to cowboys.
Though early cities and their surrounding farmland are far more dense/efficient than normal for the tech level (for reasons beyond the scope of this question). So you will also have livestock birds which nest in a set location, and feed off the surrounding wilderness, or use a nesting wagon/sled pulled by animals often with little human involvement.
This is a technologically primitive civilization which figured out mendelian genetics way earlier, but takes hundreds of thousands of years to start building cities. So there's a lot of time for selective breeding to do its work. The timeline of this setting also means you can consider as potential domestication targets species from up to the last half million years.
The proposed species need to be at minimum fast growing, flocking, and tasty. They should be able to be herded between feeding spots like free range grazing cattle. Ideally this means you can exploit good grazing/browsing spots that are in rough terrain and/or very far away from the nearest settlement. Being too helpless is also a downside, because while the magpies can be very loud and frightening, they can't keep a constant eye on every bird at once.
I've considered birds like peking ducks, and passenger pigeons as good candidates, but there's probably other good candidates I'm not thinking of, as well as factors I'm not considering. Still if I'm not mistaken, passenger pigeons seem like a particularly good candidate, because they would turn borderline inedible acorns into bird meat and/or eggs.
I also don't know how dramatically it changes things when you have domesticated birds that are herded around like this. Since the birds in question need to maintain their flight, until they're brought back to a settlement and being fattened up for slaughter (like grazing cattle).
Wild birds are still herded into nets and eaten, but I imagine the livestock birds probably taste better and aren't as seasonal. Also over time I'd expect that like with RL grazing livestock these domestic birds may to some extent displace wild birds in the same niche.