New Delhi, Jan. 27 -- A SPOT AT Little Friends, an independent child-care centre in Greenwich, Connecticut, is highly sought-after. Parent testimonials are glowing. The waiting list is years long. Yet the competition for enrolment may not be as fierce as the competition to buy it. Verna Esposito, who founded and runs the centre, has watched for years as private-equity firms have snapped up struggling rivals. They would gladly add her centre to their ranks. "They email, they call-they even sent me a Christmas gift," she says. "It's relentless."

As the covid-19 pandemic disrupted the child-care industry, large firms acquired small providers at low valuations. Firms run by private-equity shops now control 10-12% of America's child-care mark...