BEIRUT—Many of the protests that spread across the Muslim world on Friday, with violent mobs targeting diplomatic compounds of the U.S. and its allies started out as relatively small and restrained but rapidly grew out of control when groups of extremists riled up the crowd.

Similar scenes were repeated in parts of the Middle East, Africa and Asia, as crowds of men and women carried placards denouncing the video trailer for a purported film called "Innocence of Muslims," and directed their anger toward the U.S. and Israel....