Since your question is on air transfer but based on temperature difference, let's take a crack at how heat really transfers.
1. Conduction - where a steel pan edge gets hot even if the heat is only applied at the bottom. There is no movement of heated particles.
2. Convection - heat is transferred by actual movement of heated particles. In a room with the floor made up of fire, the air near the fire gets heated via conduction, as it is heated it expands and becomes less dense. Colder air up near the ceiling is heavier and "falls" down, causing the light hot air to rise up. And then the fallen colder air gets heated by fire and the process continues and room temperature rises.
3. Radiation - Hot particles give out infrared radiation which contains heat energy. Like when placing your hand near a hot stove you would feel heat radiating towards you.
Now, in your question, yes things try to maintain balance. But most of the transfer of heat is being done by radiation rather than actual movement of air (in case exhaust wasn't present). If exhaust is present, it shall throw the hot air out, but temperature difference would not cause the amount of air thrown out to be of any significance.
A case can be made that since hot air is less dense, the exhaust is actually throwing lesser hot air outside as compared to the case when we were in a cold room and exhaust was supposed to throw dense cold air out in the hot surroundings.
Since your question is on air transfer but based on temperature difference, let's take a crack at how heat really transfers. 1. Conduction - where a steel pan edge gets hot even if the heat is only applied at the bottom. There is no movement of heated particles. 2. Convection - heat is transferred by actual movement of heated particles. In a room with the floor made up of fire, the air near the fire gets heated via conduction, as it is heated it expands and becomes less dense. Colder air up near the ceiling is heavier and "falls" down, causing the light hot air to rise up. And then the fallen colder air gets heated by fire and the process continues and room temperature rises. 3. Radiation - Hot particles give out infrared radiation which contains heat energy. Like when placing your hand near a hot stove you would feel heat radiating towards you. Now, in your question, yes things try to maintain balance. But most of the transfer of heat is being done by radiation rather than actual movement of air (in case exhaust wasn't present). If exhaust is present, it shall throw the hot air out, but temperature difference would not cause the amount of air thrown out to be of any significance. A case can be made that since hot air is less dense, the exhaust is actually throwing lesser hot air outside as compared to the case when we were in a cold room and exhaust was supposed to throw dense cold air out in the hot surroundings.