What term is used to refer to Muslims who believe only Muhammad's family members should become caliphs?

The term used to refer to Muslims who believe that only Muhammad's family members should become caliphs is "Shi'a." This belief is rooted in the idea that leadership in the Muslim community should remain within the Prophet Muhammad's lineage, specifically through his cousin and son-in-law, Ali, and his descendants. This stance emerged in the early years following Muhammad's death, leading to a significant division within the Muslim community regarding rightful leadership and authority.

Shi'a Muslims hold that the spiritual and political leadership of the community, or ummah, should be vested in the Imams, who are considered to be divinely appointed leaders descended from Muhammad's family. Their beliefs contrast sharply with those of Sunni Muslims, who accept a broader interpretation of who is eligible to lead, emphasizing the community's choice and consensus rather than lineage.

In the context of the other groups mentioned, Harijites and Khwarijites were both early Islamic sects that arose due to disagreements over political and theological issues following the death of Muhammad, but they do not exclusively advocate for leadership based on familial ties to the Prophet.