They don't have authority over local jurisdictions, but they do over protection of the President, Vice-President, or others in order of success. Doesn't actually say anything about former Presidents:
https://uscode.house.gov/view.xhtml?req=granuleid:USC-prelim-title18-section3056A&num=0&edition=prelim
EDIT: Actually, that was only part A, here's the whole section which includes protecting former Presidents:
https://uscode.house.gov/view.xhtml?req=granuleid:USC-prelim-title18-section3056&num=0&edition=prelim