Jim has provided the link to Merklinger`s writings, which are the source many large format shooters have used to know about this topic.
Norman Koren also have a good one.
As mentioned, the depth of field is related to the aperture diameter (entrance pupil). This aperture is given by the focal lenght and the f number. (Diameter=FL/f number). So the depth of field increases as the focal lenght decreases (keeping the f number), or as the f number increases (stop down).
If you use a 300mm at f45 on a 4x5", a 6x9 and a 35mm (full format), you`ll get the very same DoF, because the diameter of the entrance pupil doesn`t vary. As mentioned, the format size is irrelevant.
But you need to take into account the print to make your calculations. The larger the print the more detail you need on the film. The DoF you get on the film must be in accordance to the print size and viewing distance.
If you are using a 300mm in all cases, you`ll get a slight advantage using a larger format, because the level of detail needed on a given print size (circle of confusion) is not as high as with a smaller format. The DoF will be slightly larger then.
The DoF is a fuzzy concept, more "mathematical" than "realistic", many people use internet DoF charts like if it were the panacea. We use to think on a few aspects of the shot, but as mentioned, it is only a part of the final image...
the print. They are intimately related. The "simple resolution" is not that simple at all.
But (I understand) the point here is what to do if you cannot find a 300mm lens for your 6x9 camera, or what to do if you get the 300 but you need to get a wider aperture (while keeping a reasonable DoF), in order to use a faster shutter speed to freeze the action. You may also need to keep the same field of view (the bird should pass into the viewfinder area!): Use a smaller format with a shorter lens.
(And the format issue; again, IMHO larger is not always better. Sometimes is better (easier, cheaper, faster, sharper, and the most important,
feasible) to shoot smaller than to shoot slightly bit larger with no reason).