I've had floods, fires, storms, disease outbreaks, and earthquakes. The floods and earthquakes were off-screen, things that affected the societies directly, but the players only indirectly. The storms and fires, I've used very directly to drive characters's actions. One player was shipwrecked by a storm, losing her mentor in the process. Another was thrown from his horse in a heavy storm, and left to travel on foot. Another player had to flee a running prairie fire, swimming a river to escape it. The disease outbreak set context, but the players learned there was a region to avoid.
I try to make sure the player has a reasonable chance to survive the disaster. The shipwreck required a number of swim rolls, but I only did it because I knew swimming was the character's forte. For the disease outbreak, they had a choice and forewarning. They opted to avoid, but could easily have taken precautions and cut right through the region. And for the prairie fire, he was on horseback, and smelled smoke and saw fleeing wildlife far enough in advance that he could get to the river. (His character was local, so I granted some knowledge of survival options.)