As it turns out, most mechanical shutters actually sweep down the sensor, as opposed to sweeping left or right. This is done partly to reduce the effect of rolling shutter (since the distance down the rectangular sensor is shorter than going side-to-side).
The shutter is composed of a “first curtain” that drops down (exposing the sensor) as the exposure starts, and a “second curtain” that drops down (obscuring the sensor) as the exposure ends. For a fast enough shutter speed (faster than about 1/200 second), the second curtain starts to close the shutter before the first curtain has reached the bottom. For a very fast shutter speed, like 1/2000 second), the action of the two curtains makes a slit that travels across the sensor. Here is a rough depiction of a fast shutter in the middle of traversing the sensor.
If there is motion in the scene you are photographing during the time it takes for the shutter to traverse the sensor, especially sideways motion, the top of the frame will see a slightly different scene that the bottom of the frame. A good example of this is what happens when photographing a fast-moving car. Here is what could happen.
Every bit of the car is sharp—the fast shutter froze its motion. But the car moved while the shutter traversed the sensor. (This shutter mechanism went from the bottom of the sensor to the top. Do you see why?)