If inside the same space:

Let t be a real number running from 0 to a(ge of universe). We contract space to a point inside itself by sending each vector v at time t to (a-t)·v. So at t=0, v is wherever it should be, and at t=a we get 0 (italic to denote it is the vector 0, not the number 0) regardless of v. And in-between, it moves towards that inevitable 0.

This does not "jump" (it is continuous), but from an external view, v does move with speed |v|/a (with |v| the distance of v from 0) all the time. So points far away move arbitrarily fast, similar to how some parts of the universe move away from us faster than the speed of light. But "locally", so if every point only observes those close to it, points have almost the same speed and direction. So within a close bubble, the rest of space moves only slowly.

Now we have the issue that the real universe is not contracting/expanding "within itself". This requires some slight fixes and makes the calculations a bit more ugly (hence why I did the above first):

One should think about the universe at each time t as its separate thing: imagine the Universe as a planar flat thing; now draw a time-scale in another dimension (so we need 4D if we do the real thing); lastly, fix a random point B ("Big Bang") at distance a from U and draw all the "rays" starting in B towards each point of the universe.

If U were a perfect circle, this gives you an actual cone, and this is indeed called the cone construction. Whatever U was though, one can now contract towards B in this cone(ish) thing we made as before.

Now that is still pretty far from what General relativity tells us, but I hope it explains how one can model such a thing.