The DX lens would have more depth of field in your example at the same aperture, because the focal length dictates the depth of field. The FX camera would allow less DOF at the same aperture, which actually gives you more control over DOF in my opinion. To get the same DOF with the 200 and 300 mm lenses, the FX lens would need to be about 1 stop different on the aperture meaning 200/2.8 on the DX and 300/4 on the FX.
This is the way I understand it.
Shoot nature with respect and don't trample it or startle its inhabitants. :)