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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Houses aren’t completely airtight. Chimneys, bathroom vent fans, leaky can light fixtures, even electrical sockets can leak air.
Think of your house full of warm air like a plastic soda bottle completely filled with water and the cap put on it. If you poke a few small holes in the bottom, they will dribble a bit but you won’t lose much water in the bottle. Now take the lid off the bottle.
Similar to opening a downstairs window, you’ve now made a path for “makeup” air. Water will stream from the holes at the bottom, and the total volume of water will start to go down. In your house, warm air will stream out through upstairs vents and fixtures, and cold air will rush in through the downstairs open window to replace the lost air upstairs.
Yes, it’s an imperfect example, largely because the difference in density between water and air is much bigger than the difference between warm air and cold air. Air mixing as others have posted does contribute to things, but a lot of the overall issue is hot air rising up and out of your house, assisted by an opening at the bottom of that warm air. The same effect is what makes tall smokestacks more efficient than just a plain opening.
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