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.

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“The outside money is important, but it’s not the whole story,” said  David Sklansky, a former federal prosecutor and Stanford Law School professor. In a forthcoming article in the Ohio State Journal of Criminal Law, Mr. Sklansky counted at least eight races since 2013 in which voters elected candidates who promised to prosecute police misconduct aggressively or send fewer people to prison.