The inner shroud (light grey) is fixed to the barrels at each end of the gap and rotates with it. Outer shroud (dark grey) rotates in the opposite direction, but with the same speed. Slits in the shrouds allow a cartridge to be loaded or expelled whenever both slits are lined up.
In the diagram, the barrel and inner shroud are rotating counter-clockwise, and the outer shroud is rotating clockwise. In this way, the two slits line up twice per rotation. At the top of the rotation (1.), the slits line up and a cartridge (red) drops into the breach. The slits then rotate to opposite sides of the breach (2.), sealing the chamber for firing. At the bottom of the rotation (3.), the slits line up once again and the empty shell falls out. The cycle then resets itself and starts over (4. & 5.).
It is fairly straight-foreward how the barrels and inner shrouds will rotate in a Gatling gun, but making the outer shrouds counter-rotate is more difficult. It can be done with a set of planetary gears. If each outer shroud is extended back past the slits, a set of gear teeth can be placed around each one. These can then be turned by a central fixed gear placed between the barrels. As long as the number of teeth on the central gear equals the number of teeth on the planetaries, the shrouds will rotate once for every revolution of the gun.
Again, this is in its infancy and I'll refine it a bit, but I hope you can use your imagination and get the general idea.