What shifts have occurred in world migration patterns that affect racial and ethnic relations?
The answer to this depends to some degree on the time frame that your question covers.
If we are talking about changes over long periods of time, the most important change is the increase in migration from countries of the "South" to those of the "North." We see this in such things as the migration of Hispanics to the United States and of people from the Indian subcontinent to Britain. These types of migration increase racial and ethnic tensions as the rich societies struggle to assimilate the newcomers in cultural terms.
If we are talking about shorter periods of time, an important change is in what might be called "South-South" migration. More and more, people from places like the Philippines are migrating to places like the UAE to look for work. This, too, has complicated ethnic relations as societies that had been very insular are forced to deal with people who are ethnically and religiously different from themselves.