An ancient white dragon can use its breath weapon to create a blizzard in the area around it as a standard action. This creates heavy snow conditions in a 50-foot radius for 1 minute, centered on the dragon. This snow slows movement (4 squares of movement per square entered) and limits vision as fog does.
I would say it remains stationary. It sounds like it uses a instantaneous effect (breathing) to create something that is no longer under its control. Usually if an effect follows a creature, it would specify that. I get some inspiration from the obscuring mist spell that has to be cast centered on the caster, but does not follow them.