Money from billionaire investor George Soros is helping transform once-sleepy races for local prosecutors around the country, with some candidates shedding the traditional tough-on-crime tone and promising to make the justice system fairer.

Mr. Soros, a major backer of liberal causes, has contributed at least $3.8 million to political-action committees supporting candidates for district attorney in Arizona, Colorado, Florida, Georgia, Illinois, Missouri, New Mexico, Texas and Wisconsin, according to campaign filings.

“We...