It is the mostly white-collar jobs which are moving to low-tax states. Most of the employees, working with these firms are pro-GOP. The recent phenomenon of Red states turning blue has more to do with demography than inter-regional migration.
Some of these white-collar industries (such as Banking, Equity trading.etc) are heavily dominated by the pro-Democrat types. Demography is also an important part in the change, but can't ignore this factor also.