At the center of a criminal probe involving Hillary Clinton’s handling of classified information is a series of emails between American diplomats in Islamabad and their superiors in Washington about whether to oppose specific drone strikes in Pakistan.

The 2011 and 2012 emails were sent via the “low side’’—government slang for a computer system for unclassified matters—as part of a secret arrangement that gave the State Department more of a voice in whether a Central Intelligence Agency drone strike went ahead, according to...