The triggered ability’s control-change effect has no duration. Each Dragon you gain control of this way will remain under your control until the game ends, that permanent leaves the battlefield, or some other effect changes its controller.

When the triggered ability resolves, you’ll gain control of all Dragons, including the Dragons you already control (such as Karrthus itself). Usually this will have no appreciable effect, though it will override any previous control-change abilities (such as if you had gained control of one of those Dragons only for the rest of the turn). Then you’ll untap all Dragons, including the ones you already controlled and Karrthus itself (if it had somehow managed to become tapped by then).