Air will always move from an area of high pressure to an area of low pressure to attempt to reach equilibrium. The difference of pressure over distance is referred to as an air pressure gradient.
Uneven heating (hotter air has higher pressure) and uneven water vapor content (drier air has higher pressure) causes the air at different locations to have different pressures. The air will begin to move from an area of high pressure to low pressure, however the area of low pressure will move because the Earth is rotating.
Take a paper plate and put a marker in the middle. Pull the marker to the edge of the plate directly in front of you. You will get a straight line, from high pressure (center of the plate) to low pressure (edge). Now do it again only have a friend turn the plate while you do it. As you pull the marker directly toward you, the spot you were originally aiming for has moved. Once you reach the edge you will have a curved line, even though you moved the marker in a straight line. You have just demonstrated Coriolis effect.
So the air is moved by a pressure gradient in an attempt to reach equilibrium, but the air is also deflected by the Coriolis effect which prevents equilibrium from being achieved.
There are other forces in effect that will eventually allow equilibrium to be reached, however uneven heating and water vapor will start the process all over again.