How do plates diverge?

As you stated, plates move due to the convection currents. Divergent Plates are also triggered by the same method. For example, The Atlantic Ocean lies on a Divergent Plate. The Atlantic is spreading by 10 cm every year (fast in geologic terms). If the Atlantic Ocean is growing in size, it means that the Pacific ocean is getting smaller in size (of course, The Earth cannot grow in size!). The North American Plate is moving away from the Atlantic plate, hence, it is getting closer to the Pacific Plate (remember though, this means that the edge of the North American and Pacific is at a subduction zone; while the edge of the North American and Atlantic are a divergent boundary).