I think I finally understand what's happening, thanks to a few pushes in the right direction from people who managed to understand the question I was really asking instead of telling me off for trying to think for myself.
I even think pretty much everyone is going to agree with this post, how about that
First of all, if the density is the same, hot air tends to expand since the hot molecules with higher kinetic energy can easily push the surrounding cold ones away (while speeding them up and slowing down themselves, aka adiabatic cooling). This is pretty easy to see.
So now let's look at a different situation and let's say a cube of hot air is surrounded by cold air at the
same pressure (lower density inside the cube). Suppose the hot air has double the temperature (very hot air indeed) and half the density.
Without gravity, the edges of the cube only tend to change very slowly. An undecided molecule on the boundary will get twice as many pushes from the cold side, but the pushes from the hot side will be twice as energetic. There is no net tendency for the molecules to migrate in a particular direction except for random brownian motion which is a rather slow process, even for molecules traveling faster than the speed of sound. Also, the sharp temperature gradient will smooth out because of the random nature of the collisions but this, too, is a slow process and the important thing is that it does not impose a net tendency on the entire group of molecules. So the "boundary" stays more or less in place.
Now gravity is added to the mix. Apart from the pressures on the sides of the cubes (momentum per unit square of cold molecules flying towards the hot ones and vice versa), a little bit of extra pressure differential is needed to compensate for the weight of the molecules.
If you just look at a cold column of air, a pressure differential will automatically develop so that each individual molecule gets slightly more pushes from below than from above, compensating for its weight. Otherwise all molecules would just tend to fall down and the pressure gradient would automatically develop.
Of course that doesn't mean each molecule stays exactly where it is, only that there's no net tendency for all the molecules to come crashing down.
Away from the cube, the pressure at an altitude just below that of the cube is just enough to support cold air molecules above. In other words, the molecules together are on average giving just enough extra collisions to compensate for the weight of the individual molecules above.
The pressure underneath the cube must obviously be the same because otherwise a horizontal wind would arise (higher energy molecules getting less collisions from the low pressure side and therefore moving that way).
But the hot molecules need less kinetic energy to compensate for their weight, since there aren't as many of them. The kinetic energy due to temperature is already compensated for (the hot molecules are more energetic but there are less of them, so the same pressure is required to keep them in place) but the extra pressure gradient due to gravity exceeds what is needed to counteract their weight. Each hot molecule, on average, is getting slightly more than its fair portion of anti-gravity-pushes, so they all tend to be pushed up.
Now this push is one that has a definite average value (unlike the average value of the pressures on the side which cancel out to zero), so the hot molecules will transmit their extra energy to the molecules above them as well (and meanwhile keep getting more energy from below). This "information" (extra momentum in a definite direction, up) travels at the speed of sound so that very quickly the top of the cube starts moving as well.
At the top of the cube a similar situation occurs: pressure above is lower in exactly the right proportion to roughly keep cold air in place (in fact the gradient works by giving more push from below
and less push from above), so the top wants to go up just like the bottom.
So indeed, the whole pocket goes up as a whole! Apart from the diffusion at the edges that always occurs, it does more or less seem to be a collection of molecules going up, unlike the lenticular cloud and the pocket of young people outside Florida. I had posted that latter possibility as such, a possibility, not a theory, and it turned out to be mostly not the case. The actual molecules are going up, not just the distribution of molecules that happen to be hot at anyone time.
I know what you are all going to say now: "see, we were right all along, the cube acts like a single object, the macroscopic laws are correct, I'm glad you saw the light" but the point is that the phenomenon is now explained on a molecular level, which many of you said was impossible (even "nonsensical") to do.
