Do we know how the weather in the Reach compares to weather in the North during Westerosi wintertime? Presumably, southron regions are still relatively warmer than northern regions, but I don’t believe GRRM specifically addresses the issue.

Here’s what GRRM has to say:

And, by the way, does it snow in the South during the winter?

Yes, some times, in some places. The Mountains of the Moon get quite a lot of snow, the Vale and the riverlands and the west rather less, but some. King’s Landing gets snow infrequently, the Storm Lands and the Reach rarely, Oldtown and Dorne almost never.

So the Reach rarely has snow even during the winter. 

Is it likely that there is a significant portion of the peoples of the North that migrate to the southern kingdoms (especially after Westeros’ unification) with the onset of winter or does the shorter agricultural windows during the climactic cold periods Planetos calls winter especially necessitate all hands on deck for the communities to survive, adding to the benefits of the Winter Town System? Thank you.

Well, we do have an example of such a migration, in the case of Cregan Stark’s invasion of the south in 131 AC:

Nowhere in the Seven Kingdoms did the winter matter more than in the North—and the fear of such a winter had driven the Winter Wolves to gather beneath the banner of Lord Roderick Dustin and die fighting for queen Rhaenyra. But behind them came a greater army of childless and homeless men, unwed men, old men, and younger sons, under the banner of Lord Cregan Stark. They had come for a war, for adventure and plunder, and for a glorious death to spare their kin beyond the Neck one more mouth to feed.

The day after the executions, Lord Stark resigned as Hand. No man ever held the office so briefly, and few left it as gladly. He returned to the North, leaving many of his fierce Northmen behind in the south. Some wed widows in the riverlands, others sold their swords or swore them in service, and a few turned to banditry.

So it does happen, but it’s a rare phenomenon that takes both an extreme winter and a prolonged war.