Arnie, my understanding is that if you photograph from the same bird from the same spot with a full frame and a crop body with the same focal length that the d-o-f will be the same …

a in Lima, Peru

It does not affect the working AF for the lens – an f5.6 lens is f5.6 on APS-C and on FF so will not affect the focussing as described above.

When it comes to the final image, there are so many variables:
If you frame a APS-C shot the same as a FF shot from the same position, it means you will have used a wider angle lens on APS-C. So both lenses set at f5.6 the APS-C will have greater DOF.
If you use the same lens then you have to move further away with the APS-C shot to get the same framing. The greater distance gives greater DOF with APS-C.

With birds, where you are limited by how close you can get the chances are you will crop both images (APS-C and FF) to the same FOV. Sensor size then becomes irrelevant and the DOF is then dependent on pixel density.