A rout in emerging-market assets has sparked concerns that the turbulence could spread from distant corners of the world to the U.S. and elsewhere.

A sharp selloff in Brazilian stocks and its currency late last week fed into a wider retreat that hit emerging-market assets from Mexico to South Africa and rattled Italy’s weakened bond market. Many currencies of developing countries are near multiyear lows, despite a boost from strong commodity prices and solid global growth, while investor allocations to emerging-market-focused...