The criminal charges were the latest twist for We Build the Wall, which had drawn praise from top Homeland Security officials while causing controversy over its construction methods.
Mr. Kobach, a polarizing figure in Kansas politics and an ally of President Trump’s, was defeated by Roger Marshall, the preferred choice of many Washington Republicans.
In states including Michigan, Missouri and Kansas, mail voting was put to the test as some of both parties’ fiercest advocates for their bases competed for the House and Senate.
Kris Kobach is competing in a Senate primary in Kansas that has Republicans nervous, and Rashida Tlaib is hoping to hang on against a Democratic challenger in a Michigan House race.
Republicans worry that if Kris Kobach, a polarizing figure in the state, wins the nomination, it would pave the way for Democrats to capture the seat and perhaps gain control of the Senate.
Some in the G.O.P. want President Trump to endorse the opponent of Kris Kobach, who they worry could cost them a traditionally safe Senate seat. So far, the White House has declined to do so.
Mr. Kobach’s polarizing presence in a race in a reliably Republican stronghold has party leaders worried that Kansas will become a high-stakes battleground for control of the Senate.
In contests in Kentucky, South Carolina, Kansas, Maine, Colorado and Arizona, Democratic candidates raised more money in the first quarter of 2020 than their Republican opponents.