DALLAS — The Dallas County district attorney, Faith Johnson, often reminds voters that she recently won a rare murder conviction against a white police officer who shot into a car full of teenagers, killing a black 15-year-old boy.

“They couldn’t get that conviction in New York. They couldn’t get it in California. They couldn’t get it in Ohio,” Ms. Johnson, a Republican running to remain in office in November, told the mostly black crowd at a recent candidate forum at the African American Museum. “We got it here in Dallas County.”

But then her Democratic opponent took the microphone and pledged to be even tougher on the police. And he promised that if elected, he would reduce the number of Dallas County residents who end up behind bars.

“In the first 90 days, I’m going to give you a plan to end mass incarceration,” said John Creuzot, a former judge who hopes to unseat Ms. Johnson in November.