^^over time and with subsequent migrations.
When you live in fertile lands with a Mediterranean climate, surviving is easier. That's why the Neolithic began with the Natufians. There's a great video explaining how the Levant was a paradise for early men.
You can fish from the sea as well as rivers, forage for edible plants and berries almost year long, including the wonderful chestnut trees and hazelnut trees, hunt wild game, i.e. boar, rabbit, birds, and some pigs to take you through the winter and a cow or two or goats and sheep. There's pasture for those animals almost year round. In my own Lunigiana valley, much of which has the Mediterranean climate of Liguria, you can grow crops three seasons a year. The olive tree was also a great boon once the Greeks and Romans introduced it. As for dairy products, hard cheeses cause few problems, and you don't need to be able to drink the milk of the animals to survive.
Of course, despite the beautiful climate, most of Liguria is mountainous with poor soil, but, as I said, a Mediterranean climate and decent soil is a winning combination. I don't think the selective pressure would be as strong.
One other thing occurs to me as a possibility for the timing of the beginning of the sweep. Lactating all year long is not something that these domesticated animals did naturally. The cows, for example, lactated when the young were born, as do humans. It took a long time to get these animals to provide the calorie rich milk year round.
Another point which I think needs to be remembered is that