This is from 2 years ago but it is still probably the best article on the subject, beginning with the current role of Dominionist politicians like Ted Cruz and going on to a detailed synopsis of the movement’s origins, objectives, and structure.

DOMINIONISM RISING: A THEOCRATIC MOVEMENT HIDING IN PLAIN SIGHT

Meanwhile Cruz and other national pols comprise the tip of a very large, but hard to measure political iceberg. There are untold numbers of dominionist and dominionism-influenced politicians and public officials at all levels of government and who even after leaving office, shape our political discourse. Roy Moore, the elected Chief Justice of the Alabama Supreme Court, has been a rallying figure for dominionists of all stripes for the better part of two decades. Most recently, he has led efforts to exempt Alabama from federal court ordered compliance with marriage equality, citing his view of “God’s law.” Moore’s fellow Alabaman, Justice Tom Parker, has been on the court since 2004, and has employed theocratic legal theorist John Eidsmoe as his chief of staff. 15 Others at the top of recent American political life have included Sarah Palin, Michele Bachmann, Rick Perry, Mike Huckabee, 16 and Newt Gingrich. 17 Other prominent elected officials in the dominionist camp include Lt. Gov. Dan Patrick (R-TX), 18 Gov. Sam Brownback (R-KS), 19 Sen. James Lankford (R-OK),20 and Rep. Steve King (R-IA). 21

Prominent politicians’ involvement in dominionism is certainly the most visible evidence of the movement’s advances over the past half-century, but it’s not the only result. Dominionism is a story not widely or well understood. Because this is so, it is important to know what dominionism is and where it came from, so we can see it more clearly and better understand its contemporary significance.

If you understand this movement, you will understand what is really happening in the Republican Party today and why Evangelicals are so ardent in their support for such an unlikely character as serial adulterer Donald Trump. They support him for the same reason the Russians do: He will destroy the concentrated power of the United States, particularly the Federal government. For Russians this will remove the main obstacle to their expansionist goals. For Dominionists, it will allow the rise of their decentralized theocracy, centered in local gentry as the privileged class.