A lot of this seems to be a fanciful (though possibly helpful) visualization, not firmly rooted in reality. The lungs fill up like a balloon — all parts taking in air and expanding at the same time — not like a cup into which water is poured. It is not actually possible to fill them with air from the bottom up, as described.

And we do not have lung tissue in our bellies. The diaphragm divides the heart and lungs (above) and the other internal organs (below), and it attaches to the lower ribs, the sternum and the lumbar spine. The movement of the belly during breathing does not represent air coming into the belly, but rather the displacement forward and out of the internal organs that happens when the diaphragm bowls downward and the lungs expand with the inhalation of air — the other organs are somewhat displaced as the lungs expand.

I think it is useful to be aware of this movement and allow it to happen, because if the belly is held rigid, less air can be inhaled. But air is not dropping into the belly.