optics - Lens equivalent depth of field

This question is ported directly from Physics SE, because the guys there believe this is the right place for it.

In digital photography, there's the principle of lens equivalency, according to which, when you shoot with a camera which has a smaller sensor than a full frame sensor (which is the same dimension as classic photography film), you effectively reduce your field of view (relative to shooting with the same optics and a full frame sensor). The resulting crop of the optical image is associated with saying that you are shooting a full frame sensor, but a lens with a greater focal length.

So in terms of the resulting image, one should think of the lens as having a focal length of ff (the real focal length) times the crop factor CC associated with the sensor being used (C≥1). this is all well and good intuitively but then comes the 'weird' part.

It is said that if you're to think of your lens to have a focal length fC, then you must also think of the camera's f-number as being multiplied by C as well. This means that if you increase your f-number (which is proportional to the numerical aperture), then you also increase the depth of field of the camera - and surely enough, for a smaller sensor, you yield a greater depth of field (for the same field of view) - even when your actual physical f-number is the same in both cases.

How can two optical systems with the same f-number\NA, have different depths of field?

I realize that some of you might say it's related to the pixel size of the cameras, but it goes the other way around: the system with smaller pixels (which means less tolerance for the spread of the PSF) has the greater depth of field. This is best demonstrated in modern days smartphones, where for the same f-number as full frame cameras, the depths of field are worlds apart.

What is the physical explanation for this phenomenon? My best guess is that for systems with smaller sensors, one must position the camera further away from the object to capture the same field of view, such that the light rays travel a greater distance and thus become more parallel before being collected by the optics, but I'd like to know for sure.