MEXICO CITY—Mexico's war on powerful drug cartels suffered a series of setbacks on Friday, including the disappearance of a prosecutor charged with investigating this week's massacre of 72 migrants, and twin explosions that suggested cartels are trying to become proficient in car bombs.

On the same day, the U.S. State Department said it will pull out all children of its diplomatic personnel in Mexico's business capital of Monterrey. Coming after a gunbattle last week in front of an elite school that killed two private security...