With a D4 you will have a much heavier camera and much heavier lenses. You may gain in image quality but it depends greatly on your subjects. With people and sports and dog photography it is not going to make a visible difference in any prints you have made.
I shot weddings for a couple years with the D3 and the D300 and the benefit I got from the D3 was primarily the ability to shoot with the 14-24mm f2.8 lens to get the ultimate picture angle and image quality combination possible. At the telephoto end I would use the D300 to take advantage of the 1.5 crop factor using the 70-200mm f2.8 during the ceremony and the lighter and smaller Sigma 50-150mm f2.8 lens during the reception. The D3 was used with the 14-24mm f2.8 and 24-70mm f2.8 lenses for wide to normal perspective shots.
There are benefits to using top quality DX lenses or even FX lenses with a DX camera. A Nikon 24-70mm f2.8 lens on a D300 or D7100 is going to produce better pictures than the usual 16-85mm or 10-200mm DX zoom lens. Same applies to using one of the 70-200mm f2.8 lenses instead of the 70-300mm f5.6 zoom.
I have seen visible differences between images taken with the D3 and the D300 with the same wildlife subjects and using the same telephoto lenses. The D3 provided better color fidelity and captured nuances that were lost in the images from the D300 (both cameras having 12MP sensors and shooting both with 12-bit RAW capture.
Getting a D600 camera is only the start. You need to add top quality FX lenses as well.