Countries such as the U.S., Canada, Australia, Brazil, Argentina, etc., were populated by outside groups after the indigenous peoples were killed off by disease and warfare.
Countries such as the U.K., the Netherlands, France, etc., were colonial empires that extended either/both citizenship and/or educational opportunites to the distant peoples over whom they ruled and many of those peoples moved to the “homelands” as the empires contracted, seeking better employment opportunities.
Prior to the sixteenth century, when empires were built they were collections of contiguous lands that were brought under the rule of a central power. With the advances in ships and navigation following the sixteenth century, the European empires were able to extend themselves to all parts of the world where far more “different looking” people lived. They were also able to use the technology to move large numbers of people. Europeans expanded into all portions of the world, looking for land. Africans were transported to all portions of the Americas. The immense populations of India and China were each the source of recruitment by the U.K. and the U.S. for cheap labor.
When such countries as China, Persia, and Rome had expanded their empires, the logistics of moving huge numbers of peoples from one place to another (generally on foot) were too great to allow any massive mixing of peoples. (The various Jewish exiles never amounted to millions of peoples, either to Babylon or to Rome, for example.)
There were migrations of peoples, of course. The various Asian invasions that crossed Europe at the fall of the Western Roman Empire and the later Asian invasions into Europe, India, and China did bring many people. However, the people who actually participated in the migrations were from lands adjacent to the lands that were invaded and prior interbreeding meant that the invaders did not look radically different from the peoples whose lands they invaded.
Southeast Asia had a history similar to Southwest Asia and the Mediterranean in many ways, with the Chinese, Japanese, Koreans, Vietnames, Siamese, etc. playing roles similar to the Egyptians, Assyrians, Chaldeans, Persians, Greeks, and Romans. In both cases, the amount of ethnic differences among the various groups is not evident to the casual observer. It is only with the great world-crossing migrations implemented by the Europeans that one sees societies that developed multicolored appearances. And unlike the Americas, where disease depopulated the lands allowing many new people to migrate, the Southeast Asians shared most of the diseases of the Europeans and Africans, so they did not die off to allow themselves to be replaced. Instead, the European empires in Southeast Asia became mercantile ventures in which the ethnic compositions of the peoples ruled did not change.
China actually had a fleet that would have allowed the sort of colonization and massive movements of people prior to the sixteenth century. However, China “turned inward,” destroyed its fleet, and left the great ethnic mixing to the Europeans.