They just crop the FF image that the lens delivers, and increase its dof.
Depth of field remains the same. Angle of view changes because of the crop.
A 50mm lens with 6 inches DOF is a 50mm lens with 6 inches DOF, no matter if you put it on a 1Ds or an HV40. A lens is a lens is a lens. Depth-of-field is controlled by optics. Angle of view is the ONLY thing that changes between imaging formats. The difference is that on a FF sensor, more objects are now in view in the frame that are closer and farther away (blurrier objects appear in the frame), creating a higher contrast between the blurriest objects and the sharpest objects, which appears to create a shallower DOF when the DOF never changed.
Actually you're both wrong. If the 2 images were viewed at 100% on screen then the DOF would be the same, as stated, assuming the pixel density is the same. However, if the images were printed or scaled to the same size, say 12 x 8, then the DOF would be less on the cropped image as the lack of focus would be amplified by zooming into the image. DOF is very dependant on the medium on which the image is displayed.