What my friends and I do, whether it is air shows, auto racing, or wildlife is pre set everything we can,so we are not fiddling with menus, controls, or basic settings while the important things are happening. None of us use a D4, but, the principles remain the same.
What lens(es) do you use? VR is a goodness! Whatever you have, Shutter speed at least the same as the maximum focal length of your lens. 70-300mm=1/400 sec. But, VR doesn't help much at speeds above 1/500, so remember to turn it back on. Good technique, always!
Our lenses are: (2) Nikon 70-300 VR, and a Sigma 150-500 on a Canon.
My most-used setting is AF-C, Dynamic area, Central focusing point, matrix metering. Good for all sorts of situations. Center the subject, give it good surrounding space (Don't cut anything off) for later cropping, follow,(pan)let the AF do its magic, and shoot away. Don't forget to follow through! (Don't stop panning after the exposure is taken.)Burst rate is a relatively minor consideration for most photography.
Depending upon light, Aperture priority, And let the shutter speeds go where they may....unless I am shooting jets, then, it is Sports Mode, or Shutter Priority...1/500 mimimum to 1/1000 sec. The same will apply to fast large animals.
Small,fast wildlife...hummingbirds come to mind... At least 1/500 sec., and flash, if allowed. Using the high-speed synch function, shutter set at least 1/500 sec. AFC is good here,too.
Most "pros" go where the animals or birds are habituated to humans, and are not really skittish. A zoo, public park, wild animal park, etc. is a good place to practice, or a guided "safari",where the hosts know where the animals are, and will do their best to help you get some decent photos.
Get to know your equipment,and practice. Panning/focus on autos or tucks on a highway, Decide what combination of programs works best for you, then stick with it.