The easiest way I can explain it is to think about using a FF lens on a FF sensor to photograph a white wall. This creates a 35x24mm image (digital negative) that is uniformly illuminated.
You then crop this result to some smaller size. It does not matter if it is done using a crop sensor, or in post editing; you are discarding image area that contains light that was transmitted by the lens aperture. This is equivalent to having used a different lens, that would not require cropping, at a smaller aperture. 1 stop smaller for DX, 2 stops smaller for 4/3, etc (i.e. 4/3 is 1/4 the area remaining from the FF original; 1/4 light is 2 stops less).
Simultaneously, instead of having a 35x24mm image area you are left with 24x16mm for a DX size crop, or 17x13mm for a 4/3 size crop. Again, it does not matter how the crop was done. For any equivalent image output size the smaller image area has to be enlarged/magnified more... this is exactly the same as zooming into the FF image to 150% (DX) or 200% (4/3). And this greater enlargement/magnification makes any lack of sharpness more apparent, which is exactly what having less DOF means.
So if you change nothing but increase the crop/crop factor you wind up with less light and less DOF... but different images (image area remaining).
If instead you want the same image remaining you have to back up; 2x the distance if it's a 4/3 sensor, and the Inverse Square Law (ISL) dictates that due to light spread over distance 2x the distance equals the same 2 stops less light intensity.
However, greater distance (and shorter FL) has 2x the effect on DOF that aperture does; because it makes the recorded details smaller, which makes them appear sharper. This is exactly the opposite of increased magnification and results in a greater recorded DOF.
Alternatively you could use a shorter FL on the crop body, which again reduces the size at which details are recorded and increases the relative DOF. And if the same aperture # is used the total light is gain reduced due to a smaller aperture opening; i.e. f/4 = ~12mm diameter for a 50mm lens, and ~8mm diameter for a 25mm (4/3 crop). 12mm diameter is 113mm in area, and 8mm diameter is 50mm in area... 2x less area = 1/4 light = 2 stops.