The important components of these mass movements include the Malayo-Polynesian, Indo-European, Bantu, Afro-Asiatic and to a lesser extent, Sino-Tibetan and Altaic expansions.
And American expansions contiguous to the OldWorld Cultures include the Eskaleut and Na-Dene expansions clearly out of Old World origins at about the Old World Bronze Age: the Urheimat page of the Wikipedia eve mentions that East European bronzes have been found form the Aleutians and originating circa 1000 BC. The Uto-Aztecan and Proto-Chibchan (Colombian) peoples also had radiations in the period equivalent to the Old World Bronze Age, and this is also thought to be true in the Andean (Quechuan) and Arawak linguistic groups although this is less clear.
In all of these cases we are talking a primary window of activity that is about 5000 to 3000 years ago, or 3000 to 1000 BC (with some subsequent activity of subgroups). That is the reason why the religious group posted the map, they were using it as evidence for dispersal of peoples after the fall of the Tower of Babel.