Applications to some of America’s most elite business schools fell at a steeper rate this year, as universities struggled to attract international students amid changes to immigration policies and political tensions between the U.S. and China.

The declines affected some of the nation’s top-rated programs, with Harvard University, Stanford University and the Massachusetts Institute of Technology, among others, all reporting larger year-over-year drops in business-school applications. Some, such as Dartmouth College’s Tuck School...