It does, all the time. The poles would be way colder and the equators so hot as to be uninhabitable otherwise. The air inside your house in your example is like the air between the poles and the equator, rushing from the cold place (the poles) to the hot place (the equator) and unlike you example, the air then rushes back again. So you have the cold air falling down and moving to the equator and the warm air rising up and moving back to the poles. Even in your example, more cold air is made at your AC and if you stand right next to it, you will still have cold air no matter if the windows are open. That’s the poles. The cold spot.
You example does work very well because the AC is weak and tiny compared to the might of the sun, so there is no balance. In the earth, though, the net heat loss at the pole is equal to the net heat gain at the equator, which is why the heat doesn’t overwhelm the cold.
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