If hot air rises and cool air falls, how does an open window downstairs end up cooling the upstairs in winter?
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Hot fluid rises and cool fluid falls… assuming there are no other forces acting on them but gravity.
In addition to gravity, hot fluid has a higher pressure than cold fluid, so hot pushes out into cold. If this was the only factor it would eventually cause homogeny – every bit of the fluid would eventually be an identical mix of hot and cold particles.
Of course, hot and cold particles that touch each other will share heat and become two in-the-middle warm particles.
Also, most fluid has current. In the case of air we call that the wind. Winds are produced whenever anybody moves, opens doors, etc. The wind mixes up the hot and cold like stirring while cooking food.
When all the windows are closed, your home is largely sealed-off from all the outside air. This slows air flow, because no air can escape this confined area, so when pressed on it can only press back. As soon as even one window is open, this pressure can go out of the house, so air flow dramatically increases.
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