Adding to the other ideas here.

You could use a simpler analytical approach by overlapping a bunch of perlin noise functions. I've used this for pushing around gpu side particles and it looks sweet.

Yet another one is to go with a bunch of spheres that move around the level using whatever logic you want. They could even bounce off walls in their own physics world or collision layer. You can overlap their influences. I've used this for animating tree wind and it looked amazing. You can easily simulate gusts.

You can think of this as a simple way to localize influence.

You could even use a whole lot of them and combine them with perlin noise to define their volumes, which works a lot like volumetric clouds.

Happy to help further.