The deaths of Wanda Ferragamo and Gilberto Benetton this month highlighted a challenge for some of Italy’s most well-known family businesses: With the post-war entrepreneurs who built the companies leaving the stage, they need to ensure that members of the next generation, or trusted outsiders, are ready to step up to fill the void.

Some families, such as the Agnellis and the Ferreros, have already solved the succession issue with new leaders. But others are just facing that dilemma, with no clear solutions in sight. Ensuring an orderly shift is important in Italy, where family capitalism plays an outsized role in the economy and several well-known companies—most recently Gianni Versace SpA—have been sold to foreign investors.