A decision by robo-advisory pioneer Betterment LLC to halt trading during the post-Brexit rout has stoked questions about the broad control some automated investment advisers maintain over clients’ ability to get in and out of markets during volatile periods.

The New York-based robo adviser, which halted trading for about 2.5 hours during the market selloff June 24 in the aftermath of the U.K.’s June 23 vote to exit the European Union, says this power allows it to shield clients from poor trade execution and higher transaction...