The Abaddonites are the faction most loyal to the Warmaster, and the ones most willing to obey his command without question. They consider themselves led by the Despoiler himself, but if he were to die they would likely choose Falkus Kibre as the successor.
Should they win the resultant power struggle, the Black Legionary State would likely collapse into infighting under Kibre's inept (compared to Abaddon) management. The former Traitor Legions would become independent, and resume their own campaigns against the Imperium.
The Dark Spiritualists are the faction most loyal to the Chaos Gods, and see the modern Black Legion as having betrayed the Pantheon. They're discontent with Abaddon due to his acts of bringing structure and order to a state that they want to keep, well, chaotic. They are led by Kor Phaeron, and would likely choose him as the successor if Abaddon were to die.
Should they win the resultant power struggle, the Black Legion would likely fall back to its roots, before Abaddon started centralizing power.
The Warmakers are the faction most dedicated to raiding, conquering, and pillaging. They don't have much in the way of leadership, nor to they have any plans for succession of their own. They're likely to ally themselves with the Dark Spiritualists. Should the Dark Spiritualists win out in any potential power struggle after the death of the Despoiler, the Warmakers are likely to turn on their former allies and sink the Black Legion even further into anarchy and infighting.
Support for the Warmakers is not exclusive to worshippers of Khorne, as some of the followers of the other Chaos Gods also have an interest in raiding and pillaging.
Should they win the resultant power struggle, the Black Legion would likely die after trying to fight a vastly superior enemy.
The Dark Mechanicum isn't a conventional faction of the Black Legion. Where the other factions are mostly led by Chaos Space Marines, the Dark Mechanicum is led by augmented Hereteks and dark Techpriests. Their leadership is known only to themselves, but whoever leads them is likely to place Perturabo as the successor to the Warmaster, remaining careful to keep the actual power for themselves.
Should they win the resultant power struggle, the Black Legionary State would become an even more harsh place to live, run by the Mechanicum in the name of progress, efficiency, and the Dark Gods.
The Cursed Ones are the most mysterious faction within the Black Legion. Their leadership and even whom they choose as successor are hidden from the eyes of all but the Dark Gods, and even they might not know. Some are suspicious that they might even be planning to kill Abaddon.
Should they win the resultant power struggle, the outcome would likely be very bad.