Primogeniture was never a rule in the case of succession to the throne among the Mughals or many other dynasties. Though Humayun, the eldest son succeeded Babur without a dispute, there was not much of an empire at that time. Still Kamran, his just younger brother gave enough headache to Humayun that he had Kamran imprisoned and blinded. Hindall and Askari, the other two brothers also never helped Humayun much.
Akbar did not face any sibling for succession as he had none! But he had to face Hemu in the second battle of Panipat to secure his throne. Murad and Daniel, the younger sons of Akbar died before their father due to heavy use of alcohol. Still Salim fought against his father, but Akbar pardoned him as he had no other sons left! When Salim ascended the throne as Jahangir, he had to face the rebellion of his eldest son, Khusro.But Khusro was defeated, imprisoned and blinded, and later killed by Khurram, his younger brother.
When Jahangir died, not only Khusro, but his second son Parvez was also no more and again alcohol was the villain. His third son Khurram and the youngest Shahriyar fought for the throne. Shahriyar was defeated, blinded and killed. Khurram ascended the throne as ShahJahan.
When ShahJahan fell ill his four sons Dara, Shuja, Aurangzeb and Murad fought as usual and the result you know. ShahJahan was the only one among the "great" Mughals who had the misfortune to spend the last eight years of his life as the prisoner of his son. Later "lesser" emperors like Farukhsiyar, Shah Alam and Bahadur Shah II had suffered greater insults, but then the empire was only a shadow.
After the death of Aurangzeb also the story repeated. His sons Muazzam, Azzim and Khambaksh fought each other and the victorious Muazzam ascended the throne as Bahadur Shah, in his old age. After him, his son Jahandar Shah also came to the throne similarly. But after Aurangzeb, the Mughals had entered their period of decline.