Extratropical cyclones (a kind of low pressure system found in mid latitudes) are actually related to the movement of a cold front and a warm front, as part of a frontal wave. They drive cold and warm air masses to move.

Your claim that "air moves from warm object to a cold object" might be the root of your misunderstanding. That claim isn't true. It is true that heat will move from a warmer object to a cooler object through conduction and radiation. But that is not related to whether the air itself will move. A local effect called a "sea breeze" demonstrates the opposite scenario. A hot beach creates a warm, rising column of air (and a local area of low pressure), bringing in cooler air from the ocean to replace it.

You have conflated the transfer of heat with the movement of air and air masses.