This article is part of our Design special section about new interpretations of antique design styles.

Tabloid newspapers and television dramas have sharpened our appetites for horror stories about family businesses, of Machiavellian parents seated on the thrones of their capitalist empires manipulating their power-mad children like chess pawns.

While there has yet to be a legacy design company that could prop up a TV show like “Succession,” many grapple with the transfer of authority to younger generations. Those who inherit the mantle must face a digital, environmentally challenged, globally knitted future, the likes of which their forebears could hardly have envisioned.

Ligne Roset

In March, Antoine and Olivier Roset were named co-chief executives of Roset SAS, a parent company of the French furniture brand Ligne Roset. Founded in 1860 by an earlier Antoine Roset, Ligne Roset began as a producer of walking sticks and umbrellas. Its current leaders are first cousins and great-great-grandsons of the eponymous founder.

Antoine, 43, joined in 2006 as an executive vice president overseeing Ligne Roset’s North American division. Olivier, 42, arrived two years later as the director of finance and general director. Both now work at the company’s headquarters in Briord, France.

Ligne Roset has long collaborated with emerging designers, and the pair considers putting money into equipment to support new design an essential component for moving forward. “It is key for a manufacturer who wants to be at the forefront of design and development,” Antoine said. Olivier pointed out that a company can give carte blanche to visionary designers only if it is equipped with the technology to produce what they dream up.

In collaboration with a California biotechnology company called MycoWorks, the cousins are developing a new type of vegan leather made from mushrooms. Other companies are experimenting with alternatives to hides, they noted, but Ligne Roset wants to be the first to have a sustainable vegan leather on permanent offer in their products.

At the same time, the Rosets are revisiting furniture perennials, like Togo, a low, cocoon-like sofa designed 50 years ago by Michel Ducaroy. There is even a podcast about its development.

Maison Leleu

Alexia Leleu, 38, wants to return her family’s business to its former glory, which was abruptly interrupted 50 years ago. The Paris design house, founded in 1910 by her great-grandfather Jules Leleu, was known for limited-edition furniture, rugs and lighting. Commissioned in 1969 by Mohammad Reza Pahlavi, the shah of Iran, to create an installation for the 2,500th anniversary of the founding of the Persian Empire, Maison Leleu spent three years making 51 tents for the event. But the shah never paid the bill, supposedly because of the brewing revolution in Iran, and the business was forced to close in 1973 to avoid bankruptcy.

Six years ago, Ms. Leleu, who originally worked in the pharmaceutical industry, discovered that a former secretary at Maison Leleu, Françoise Siriex, had kept a large library of drawings of the design house’s furniture and lighting, color illustrations of fabrics, wallpapers and rugs, and photographs.

“At this moment, I decided to bring Maison Leleu back to life,” she said. Having studied art and furniture history at Ecole Boulle, the Parisian college of fine and applied arts, she wanted to honor her family’s legacy, but she did not want to simply reissue old pieces.

Ms. Leleu faced some hurdles (“It wasn’t easy to convince 40 rights holders to give me the chance to continue the family adventure,” she said). She is personally financing the business. “Naturally, I opened the doors to other family members who wanted to join me in writing this new chapter,” she said. “However, no one at the time wanted to take such a risk.” To diversify, Ms. Leleu is studying to be an interior designer “and offer a global package,” she said, and she released her own line of furniture and rugs. “It is the next step. I’m a very confident person. I trust my DNA.”

Loretta Caponi

Guido Caponi, 35, is the chief operating officer of Loretta Caponi, a Florentine atelier founded by and named for his grandmother. (His mother, Lucia Caponi, is chief executive.)

Since 1967, the company has produced exquisite hand-embroidered household linens and lingerie. Mr. Caponi studied law at the University of Florence and took a gap year to work the region’s vineyards and farmhouses, but he also spent a lot of time in the company’s archives, intending to assume a role in the business one day. It has always had “something very special, something unique inside,” he said. “What I’m trying to do is just let the other people see what I see.”

Mr. Caponi is looking into the distance, too. He said he hopes to make Loretta Caponi more international, digitize certain internal operations and expand production without altering the textiles’ identity and quality. And while these maneuvers will require additional artisans, for now, none of them live farther than 80 kilometers (about 50 miles) from Florence, and the work is still all done by hand. The company now has more than 50 dealers worldwide, and in June, it added its first shop-in-shop at Harrods in London.

“Our next dream would be to open a store in the U.S., for sure,” Mr. Caponi said. “But it will take time to do so.” For now, Loretta Caponi is featuring its linens on Abask, a new website by the founder of the clothing retailer Matches.

Wengler

Karen Wengler, 38, is reviving her family’s rattan business in Copenhagen, which has lain dormant for decades. Founded by her German-born great-great-grandfather Robert Wengler in 1850, the company originally sold housewares. The development of leisure culture at the end of the 19th century opened a market for backyard, beach and resort furniture. “They didn’t get tans at the time so they sat under umbrellas on wicker furniture,” Ms. Wengler said.

Robert Wengler went on to make pieces for the Danish royal family and Czar Nicholas II of Russia, as well as ordinary folk. The company “perfected the way wicker furniture was produced and were considered the standard,” Ms. Wengler said. As rattan evolved with the times, it worked with Danish modernists like Arne Jacobsen and Nanna Ditzel. (Ms. Ditzel’s 1959 Egg chair, designed with her husband, Jorgen Ditzel, became one of its most-loved pieces.) But it floundered on the lack of a successor when Ms. Wengler’s grandfather chose to pursue a medical career rather than take over the family operation.

Ms. Wengler mixes the skills of her forebears. She holds a master’s degree in business and began her career doing product development at pharmaceutical and medical technology companies. In partnership with the Danish-American company Form Portfolios she has revived several Wengler pieces, including two sold by CB2: a lounge chair designed in 1945 and a chaise from 1955. Additional pieces from the archives will be available later this year through the Scandinavian lifestyle brand Dansk.

Etel

An ambassador of Brazilian design, the São Paulo company, founded by Etel Carmona in 1985, not only has created new furniture pieces but also helped to preserve the country’s cultural history. In 1995, it made an enormous impression when it introduced 50 pieces at the International Contemporary Furniture Fair, in New York, and didn’t look back.

Lissa Carmona, Etel’s daughter, had been trained in finance, but changed course, immersed herself in design and went on to become the company’s chief executive in 2008. In addition to pieces by her mother, the collection now includes works by Oscar Niemeyer, Oswaldo Bratke and Jorge Zalszupin. Last year, Ms. Carmona honored Mr. Zalszupin, a Polish-Brazilian designer and architect, by creating a museum of his work in his former residence in São Paulo.