I could swear I knew the answer in grade 5 and no longer remember but like at what point is it one step to the right and no wind and suddenly wind a step further. Like rain starts from the clouds (and I assume thunder too) but what about wind.
According to the link in the top comment (sorry for caps, copied it out of the image): COOL AIR TRAVELS TOWARDS WARM AIR OVER LAND.
So the warm air does push, but it pushes upwards (rises). Cool, low pressure air, then fills the void.
This clarification brought to you by another non-expert in the field. Ha