Institutional investors for more than a decade have sought to get their own slice of private-equity’s deal-making pie. But just as firms offer up more servings, some investors are losing their appetite.

For years, private-equity firms have offered their fund investors pieces of certain deals that they back, as a way to attract and retain these investors and diversify the equity risk associated with such deals. Firms typically don’t charge investors fees on such co-investments or take a share of the profits from them, which…