Actually the two largest migrations in the last century were south to north. It wasn't the invention of air conditioning that reversed that trend, it was the enforcement of civil rights.
Life changed immensely in the 20th century as air conditioning and refrigeration systems became more efficient, controllable, and even mobile. No longer dependent on the weather for work or play, humans truly made the environment adapt to their needs. Climate control became so reliable and affordable it grew from an invisible luxury to a common necessity. By the end of the century, nearly 70 percent of U.S. households had air conditioning. Now people can live and work in glassed-in or windowless buildings, in porchless houses, or in the warmest and most humid places. In the United States alone, air conditioning reversed a century-long pattern of migration out of the southern cities.