To add an example of large-scale transportation of plants, the plant known as the black bean (
Castanospermum austral) was native to a small area of far northern Australia, Cape York Peninsula. Aboriginal people traded or otherwise carried the seeds more than halfway down the eastern seaboard, and planted the trees in all sorts of locations where they could come back and harvest them later. The black bean produces large amounts of seeds (seasonally), and was used for this purpose. Other kinds of plants were moved around, too. These kind of processes go far beyond "gathering". See the link
here for more details of this example.