While I can't answer your primary question or comment on Christianity in general, there is a similar question which has pervaded Islamic theology and practice for the religion's entire history. You may have heard of the two main denominations of Islam, Sunni and Shia. The split and the main line along which the two denominations are divided has to do with the question of Muhammad's successor after his death. Sunnis back the Ummah (Muhammad's core following, mostly those who had been with him from when he first came to Medina) in its authority to designate the next caliph; they designated Abu Bakr, one of Muhammad's earliest followers. Shiites back Ali, Muhammad's cousin and son-in-law, who they believe was nominated by Muhammad prior to his death. Sunnis held majority, as they still do today, and elected Abu Bakr the first caliph, but though Sunnis held the political advantage for the first several generations of the Islamic empire, the Shiites continued to organize in opposition.

Shiites maintained that divine truth and right to leadership of the caliphate is passed on through Muhammad's relatives and descendents. Thus, the first three of Muhammad's successors were not recognized as legitimate by Shiites, but the fourth, who was Ali and therefore a direct relative of Muhammad (and, again, Muhammad's chosen successor according to the Shiite view), was. Ali ruled from 656 to 661, during which time the First Fitna was fought over related questions of succession (as well as more tangible concerns). In 661 Ali was assassinated just as his predecessor had been. During this period also appeared the Kharijites, a separatist group once again related to the question of succession and arbitration as a means of choosing a caliph. They proved a thorn in both mainstream denominations' backsides and organized assassination and insurgency against both the Rashidun and Umayyad.

Ali's son, Husan, inherited the Caliphate after Ali's death, but his power was quickly subverted and the Sunni Umayyad Dynasty came into possession of the Caliphate. After the collapse of the Caliphate tradition, independent Muslim kingdoms and dynasties emerged and Sunnis and Shiites continued their dispute at the head of their own individual polities. The rest of Islamic history is largely a story of their continued division, and it all started over the question of who was the rightful successor to Muhammad.