Perhaps you would explain the change in focal length in the wrong direction? 50mm on FF is the same FOV as a 30mm (nearest equivalent) on a 1.6 crop, the aperture values would not change, DOF might depending on conditions
I don't think I understand. Yes, 50mm on FF shows approximately the same field as 30mm on 1.6x APS-C. You didn't state the aperture values, so I don't know what you mean when you write that they don't change. In relation to what? Focal length and aperture (as in entrance pupil) are independent quantities, and the f-number is their ratio.
Let's put it this (trivial) way. 50mm/2.8 on 1.6x APS-C is exactly
the same as 50mm/2.8 on FF, where only the central 1/1.6 = 62.5% (in linear dimension) of the frame is used.
With shorter focal length, you compress solid angle onto a smaller part of the detector; so with 31.25mm you compress the same field on the APS-C as the 50mm does on the FF. If we assume that the entrance pupil is the same, that is, the same number of photons per solid angle is captured for both lenses, then that implies that the photon density at the detector must be higher for the 30mm lens (because the same number of photons were compressed onto a 1.6^2 smaller area). That is, the f-number is sqrt(1.6^2) = 1.6 smaller.
I'm sorry if my explanations are confusing, I'm sure there is a good web page somewhere which explains things better.