Dozens of iPhone owners are taking Apple Inc. to court over its disclosure that it slowed down old phones to preserve battery life, in what could become one of the biggest legal challenges involving the company’s smartphone since its 2007 debut.

Some five dozen iPhone customers have filed at least 59 separate lawsuits since December accusing Apple of slowing their phones to spur people to buy new iPhones, according to court records. Apple said in December that its software update introduced at the start of 2017 reduced the...