How is shape casting done in physics engines

Question

I've read about a technique called shape casting, used to check for all intersections between two discreet time steps, for continuous collision detection. I wonder if this is done by sweeping a shape? A sphere can be swept to become a capsule. A box can be swept to become an elongated box. A capsule, I imagine, can be swept to become a Minkowski sum of a plane and a sphere, i.e. rounded plane. But then there are more complex shapes. Is it really possible to sweep a complex shape like a convex hull? If not, then how would one do shape casting for this types of shapes?