I find this particular question very interesting...But even more so, the responses to date!
There have been responses which had no relevence, to some which bordered on the edge of contemporay WC thinking...
Allow me to begin my response with an analogy/example.
When you walk, do you step forward with one leg out in front of your CM, then try to catch up with it with the CM? And then repeat this process, over and over?
I don't think so, as such a gait would encourage people to walk on the other side of the street from you...
Rather- don't we tend to project our CM forward, then allow a leg/foot to get out in front, absorb the impact, carry the CM over that point of balance/contact, while the next leg/foot swings forward to repeat that performance? In this way, we control our speed, direction, and adaption to terrain and obstacles in our path.
Many of the responses to this thread proposed classic dogma, tipping, turning etc, then position the CM. I would agree that some of this is required, as skills are being developed, muscle memory established, and movements formed. But I sure hope you don't expect this to be your path to upper level or elite skiing! If so- good luck in getting there!
Talk to elite/WC athletes, and you'll hear a very different scenario. More like what Bob Barnes has often promoted- "I want to go there", now do what must be done for that to happen. And almost invariably- it leads with the CM, not the legs/feet. And once the CM is moving where they want it to go, they move/position their legs/feet to support that mission.
An example of this concept. Imagine yourself running down a boulder strewn creek bed. Do you move your legs/feet from boulder to boulder, causing your CM to move erratically, negatively affecting both your balance and the continuity of your chosen path? Or do you attempt to maintain a relatively calm CM, controlling it's movements, while allowing a much more active lower body to smooth out the path?
There has been some discussion as to whether the CM can be held still enough to provide a stablizing influence. Utilizing Newton's Law which states that "an object in motion will remain in that motion until acted upon by an outside influence"(abbreviated), it can act as a stabilizing influence. The more consistently it moves, (in direction, speed, and elevation) the greater the influence and the greater the freedom of the lower body to support it. And the actions of the lower body will be precisely timed and executed.
But the key to utilizing this concept is the ability to identify and isolate your CM. If you can not identify your CM (core) accurately, and envision the path you want it to take, then you will continue to ski by making arbitrary movements of the legs/feet, hoping your CM remains somewhere within the balancing zone.
This also begins to tie in the previous discussion regarding cross over/ cross under. In fact, at the elite levels, 99% of all turns are cross overs. Regardless of the radius, it is the timing, duration, and intensity of the movements which make it so. Obviously, it happens quicker in the shorter radius', as a relaxation/flexion of the legs allows the CM to remain unfettered as it crosses over the skis. In a longer radius, with greater time and distance covered, it occurs more visibly, with more extension usually involved.
The 1% which are cross unders are purely reactionary corrections, and have a distinct displacement of the skis from one side of the CM to the other.
Hopefully, my comments will spark further thought (outside the box of traditional dogma), and help expand the understanding of what elite level skiers are concerned with, and how they go about harnessing the natural energy of their skis, the hill, and their own bodies.