Atoms weren't really taken for granted as existing until the Scientific Revolution (and were not really a major part of the understanding of the world until the late chemical revolution, and not really understood until around the early 20th century), so that's a long, long time after anyone believed the world was flat (it was known to be round in antiquity).

As it is, your query really comprises a huge span of possible times and places and people. But let's be easy and just talk about Aristotle, both because his approach to the world is pretty comprehensive, but it is utterly non-atomic, and, helpfully for us, he wrote a whole book about meteorology (and we have it!). Aristotle, like lots of people in his day, believed the world was composed of matter that took on the qualities of one of four elements: fire, air, water, and earth. Specifically they tend to cluster in the world in that order — fire is very high up, but then comes the air we breathe, then lakes and ocean, and under that earth. So there is a "layer" of air in Aristotle's model, and this is where things like weather and wind take place.

Why does wind blow? Aristotle has a lot of discussions of different types of winds and weather in his Meteorologica, but the most basic is that the heavens are rotating (the Earth is stationary in Aristotle's model), and this rotation causes the winds to move around. But if that were true, then the winds would only blow in really straightforward, common directions. Aristotle notes, correctly, that air can take on different temperatures (get fire associated with it, in his model), in part because the Sun warms the Earth, and that causes all sorts of variations (winds rise, fall, mix, etc.). So that's enough of a fudge factor by itself (heavens are moving, airs of different temperatures move differently — hot air goes up, cool air goes down) to allow for a lot of complexity. Aristotle is adamant that wind is pretty complex, but is ultimately related to lots of independent actions of air being heated or cooled and moved around by the heavens. (You can read more here if you want — his logic is pretty easy to follow.)

Like much of Aristotle's view of the world, you can kind of see how this makes sense, even if it's not very useful. (Aristotle does not care — his goal is not to be useful, but to be true!)

Anyway. If you're asking what wind "is," like, what is it made of, that's easy for Aristotle: it's air, one of the four elements that makes up everything. "But what's air" is not really a valid question in Aristotlean cosmology, because air is by definition just air (anymore than asking "so what are quarks made of?" is a valid question under present quantum mechanics — quarks are just quarks, they are themselves, they are not made of anything because they are fundamental).