MWAHAHAHAHAHAHAHA! I have it!
I think I do, anyways. I was able to pull up the following pdf off of Google Scholar about the behavior and history of the saiga antelope:
http://www.ubss.org.uk/resources/proceedings/vol18/UBSS_Proc_18_1_74-80.pdf
It's actually the most detailed thing I've read thus far on them. Knowing more about their behavior has shed some light as to how they might become incorporated into society in North America as an early replacement for sheep. Apparently they mate in winter, and do not have a very set migratory range, following prevailing winds and migrating into a variety of different habitats. They have a preferred habitat, which is dry, arid steppe, often subarctic or even arctic (as seen in the North American variety), but in times of extreme summer drought, will look elsewhere, in scrublands, and even open woodlands. Calves are born in spring, and they migrate into more southerly pastures during the winter to escape the snow.
I did a little research on how reindeer were domesticated, and it appears that they were herded into pits and butchered or kept over time. I therefore see no reason why saiga could not become the staple food of a population of marginalized humans (probably Denisovans) in Alaska, who would herd them and keep them into similar pit pens and keep them from performing their yearly migrations for winter storage, which could over a significant amount of time, with calves being birthed in these pens and imprinted by humans, result in domestication.
This may occur in a few areas, however. A group in more forested areas of Alaska might want to keep saiga for medicinal purposes, or possibly for the taste of their meat (which apparently is another reason they've been poached over the centuries), perhaps when the spring rains are poor and the saiga stay where they know the water is, resulting in similar imprinting of calves in captivity by hunter-gatherers. If this kind of trapping is done successfully over generations, with those imprinted calves being released that the hunter-gatherer bands could not keep in pens after they moved, populations that do not mind human presence would ever so slowly, but surely develop, eventually paving the way for herding.
These human herded saiga would be in a better position to survive once the climate begins to change, as by this point in time, humans, and not the saiga themselves, would be leading the migrations. I imagine the initial stage would happen about 120-130,000 years ago in my timeline, with the final process leading to the full domestication about 10-11,000 years ago at the end of the Last Glacial Period. Once the wild populations are extinct, the human herded ones will not have wild counterparts with which to breed freely, and more selective breeding will be able to take place. The saiga would then spread out of these isolated pockets with the humans that domesticated them, across modern Canada, and south into the modern United States, being bred for its wool, its meat, and its milk.
Maybe as a draft animal, but
Camelops and
Hemiauchenia, who would both have adapted to more than 100,000 years of human presence before the arrival of modern humans, would definitely be the best bet in this department.