Kids at those ages are never more than 10' from you. Sure there are playground run arounds, birthday parties and all that where the kids are farther away from you, but I don't think a 24-120 or a 28-300 is the right choice. A one year old and a four year old are easily captured in all light, motion frozen, with the versatility of a fast 50mm and the low light capability of anything from a D5100 through the D800. Indoors, even with flash and all its attendant considerations, the 24-120 and the 28-300 just can't keep up with a fast 50mm.
The 28-300 VRII is terrific, but it has its limitations. Edge distortion in some situations can be startling. Nevertheless, a couple of Nikonians in particular are getting superb results out of the 28-300, but certainly take the time to frame and compose carefully, work the lens to its strengths, and spend some carefuly considered time in their post-processing software. The 28-300 focuses moderately fast, but won't be able to keep up with busy, energetic kids horsing around in the yard in many situations. Its slower maximum aperture at full zoom slows AF down a bit more (even considering the slower minimum stop AF systems in the D600 and D800)
The 24-120 is faster lens all around than the 28-300. Optically, the 24-120 provides consistently better results than the 28-300, but with the proviso that you have to look r-e-a-l-l-y closely to see the differences in many photo comparisons. Once again though, I think its appropriate to ask yourself if the 50mm f/1.4 AF-S would be a faster and more versatile choice for photographing your one and four year olds.
The apparent versatility of a 24-120 or 28-300 zoom often does not apply in practice, which is a bit counter-intuitive I suppose. The reason is that aiming, zooming, reacquiring focus and pressing the shutter button all take time. As soon as the light drops to a certain level outside or inside, you either have to bring out the flash (which a lot of kids don't like after one or two shots), or do the smart thing and mount a 50mm f/1.4. Handling a longer, bulkier zoom around busy kids can be problematic in close quarters. The close focusing distances of the 24-120 and the 28-300 (1.5' and 1.6' respectively - about 0.5m) are quite good, but AF accuracy drops somewhat at their close-focus limits. Acceptable, but not great.
By contrast, using a fast prime like the 50mm f/1.4, you just aim, acquire focus and shoot. The more you use the lens, the more you begin physically positioning yourself at the best possible location to make the shot. The 50mm prime lens is less obtrusive than a much bigger zoom lens which can put some kids right off. You don't need a flash indoors as long as there's enough light in the room for the kids to safely play or do whatever they're doing. The sharpness, bokeh and speed of the 50mm f/1.4 AF-S prime lens is not matched by the 24-120 or the 28-300. Indoors, the fast 50mm shines, and at shorter focusing distance (the same 1.5' as the 24-120) its AF is more accurate will help make many shots you'd otherwise not get. The faster maximum aperture means you'll always be able to use a faster shutter speed to freeze the kids in action. The D600 with a 50mm f/1.4 mounted is a very fast photography rig which will help enable you to draw the camera, aim and shoot while a photographer right next to you is still trying to frame and lock focus using the slower zoom.