You get more DoF at the same f-number because you're using a shorter focal length, and thus have a narrower entrance pupil at the same f-number, to get the same AoV.But You get more DOF at the same f-stop if You use an APSC. At all there is no advantage to neither FF nor APSC:
Tested it with a DOF calculator: same pixel count (eg 24mpx), same AOV, same picture size, same distance. Both cameras set to the DLA, You get the same DOF.
But in the case (rain forest) in the comments above, it's being proposed to use the same focal length with a crop sensor to get a narrower AoV, in which case the higher enlargement ratio means the APS-C camera will have shallower DoF.
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