The simplest answer is from Gale Tattersall, DP of the TV show HOUSE : a larger sensor requires a longer lens to achieve the same field of view. The longer the lens the less DOF. (aperture staying the same of course)
Which aperture: the f-stop or the physical one?
If it is the former, start with the 200/2 on crop. How do you get more DOF with 320mm on FF?
Mr Tattersall was over simplifying.
How do you get more DOF with a FF than a 200/f2 on a crop camera. Easy, use a lens with a smaller physical aperture than 100mm, a 100-400 f4-5.6 would do the job.
If you wanted equivalency, i.e close to identical images from both cameras from the same spot you'd need to do this
- Crop camera: 200mm, f2, 100iso
- FF camera: 320mm, f3.2, 200iso
Obviously a 300 f2.8 on a ff camera outperforms a 200 f2 on a crop camera with regards thin dof, it also makes smoother and much more blurred backgrounds. People who espouse the 200 f2 as having a "unique look", are missing the point, a 300 f2.8 demonstrates all the "unique" qualities of the 200 f2, isolation, big background blur, narrow dof, only it does them all "better".