The LA Times offers this must-read extensive report. A snippet:

A spokesman for the Republican National Committee, Danny Diaz, said the party was closely monitoring the cases across the country but was not directly involved in coordinating strategy.
Still, the party has advised at least one state on election law. In a March e-mail obtained by The Times, the Republican National Committee’s deputy counsel advised an Arizona official that the state was free to require proof of U.S. citizenship for newly registered voters, as the new state law mandates.
The lawyer wrote that the state did not have to follow an opinion to the contrary written by the U.S. Election Assistance Commission, the panel created by the 2002 Help America Vote Act to guide the administration of federal elections.

See also this post by Bob Bauer on Republican efforts on the national level to require voter i.d.
Back in February I wrote this column for Slate: “Fraud Reform? How Efforts to ID Voting Problems Have Become a Partisan Mess.” Things have only gotten worse since I wrote that piece.