However, you are correct that heavyweights don't and have not typically "moved like that". Even those who were relatively fast and defensively gifted (e.g. a young Tyson) didn't move as poetically.
He moved like a boxer of a much lower weight class and, beyond even that, he moved like Ali. Graceful. Fluid. Quick. Rhythmic. Instinctive. Check out some of his highlights on YouTube. There are entire sequences during which he defied opponents to hit him through head/upper body movement alone. Hands are low and movement is so fast and well-timed that it seems to predict his opponent's next move. It's as if he is training his opponent to punch wherever he moves his head, and his opponent is reacting to him vs. the other way around.
And, this is to say nothing of his footwork. He glided without effort and the mechansim by which he changed direction was almost imperceptible. It seemed that he was being guided on a wire versus using his legs. Back then, it was amazing to see a man his size (or any) move like that, and it would be just as much so today.