Field of view is a function of focal length, frame size and distance.

If we assume we are dealing with one frame size only, frame size drops out of the equation as a constant, and field of view is a function of focal length and distance only.

Focal length and distance combined are magnification.

Now it is not true that DOF on film changes with field of view. When magnification and f-stop are constant, DOF will be too, no matter how large or small the frame.

Frame size enters the equation when you want to compare different frame sizes.

Let's keep the distance constant (for simplicity's sake). Then you need a shorter lens to fill a smaller frame the same way you fill a larger frame with a longer lens. That means magnification is smaller, DOF larger on the smaller format film.

When you want to compare same size images, from the same viewing distance (or smaller images from a shorter viewing distance, such that the apparent size is the same), magnification will be the same again, and so will DOF (there are a few side effects that will change the apparent DOF, but i'll ignore those for now).

Or (in short): the field of view is unimportant. Any difference you see when changing frame size is that in frame size.

There will not (!) be a large difference in DOF at all.

The thing that is correct is that unsharpness, the fore- and background blur, increases more rapidly when longer lenses are used. So there is a visible effect of focal length.

But that isoutsidethe DOF. DOF itself is still the same, as long as magnification and f-stop are the same.