They either fall off the plant and grow nearby or are swept away by water. Those swept away by water can end up far from the parent plant. Many tiny, wind blown seeds reached the pyroclastic flows, but the first successful plants were lupines. This happened even though lupine seeds are slightly larger and less easily transported. Nutrients are not available on the pumice plain.
Lupine seeds are more successful because they carry plenty of food in their seed coat.
This stored food provides nourishment for vigorous sprouting and rapid root growth. On the dry and dusty pumice plain, only a few seedlings managed to survive. Those were the seedlings whose roots could grow through the dry surface to the damp soil below.