The Manhattan district attorney is sending $38 million taken in civil forfeiture from international banks in New York to cities and counties across the country to pay for testing 56,000 rape kits that have been collecting dust in police storage rooms for years.

Making good on a promise he made last year, the district attorney, Cyrus R. Vance Jr., unveiled a list of law-enforcement agencies in 20 states that will receive grants from his office to help eliminate backlogs of the evidence kits, which contain DNA samples taken from victims’ bodies after sexual attacks.

Mr. Vance made his announcement at an emotional and celebratory event in the New York medical examiner’s office, where a decade ago the city’s push to test all untested rape evidence led to 49 successful rape prosecutions. He was joined by Vice President Joseph R. Biden Jr. and Attorney General Loretta E. Lynch, who announced a parallel effort by the federal government.

Ms. Lynch said the Justice Department was contributing an additional $41 million in grants to the cause. Those grants, approved by Congress last fall, will be used to test 13,500 kits in 20 additional jurisdictions, she said.