It would depend on the river. You don't want the Mirage Drive near rapids or eel grass in shallow water, but you can pull the drive and paddle if you need to. Current is more complex, but you should be able to easily handle anything a proficient paddler can, barring restrictions already noted.
Moving water, per se, is not a problem. Keep in mind, the Adventure does not turn on a dime like some of the other kayaks do. Downstream you need to know your boat and plan your turns in advance. Still, the Adventure can still get into some pretty skinny water, even with Turbofins:
Technique is a big factor in where you can go and what you can accomplish.
