WASHINGTON—The sudden outbreak of large protests sparked by the killing of George Floyd on May 25 at the hands of Minneapolis police came at the same time mayors across the country were working to implement more aggressive testing and tracking programs to combat the coronavirus pandemic, complicating their efforts.

Public-health doctors say the federal government hasn’t played a leading role in urging broader testing, so mayors and local officials are increasingly thrust into the breach. Scores of them are guided by leading…

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