On a warm front, warm air pushes on the colder air ahead of it. As the warm, moist air is more buoyant than cold air, it is pushed up and over the cold air. The cold air forms a wedge, which is slowly pushed away by the warm air. More often than not, a warm front produces relatively shallow convection and mostly moderate rain showers. In some cases, thunderstorms also form on the warm front.
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Warm air is pushed over the cold air, forming a wedge of cold air between the ground and the warm air aloft. Typically, the wedge angle is very low and the warm front produces little or no convection. In some cases, on more robust warm fronts, convection does occur and forms rain showers and, more rarely, thunderstorms.
Interesting fact: in Tornado alley, thunderstorms on warm fronts are prolific producers of tornadoes.
There is a third, much more rare type of fronts, called a dryline. Along this type of front, air temperature does not change much, but humidity (moisture content of the air) changes a lot. Drylines form near mountain ranges, and under the right conditions, also initiate convection.