They did this in Middle Ages Europe as well. Order people who were too close to power to leave for pilgrimmage to either Rome, or more dangerously, Jerusalem and the Holy Land. Very convenient way to get rid of royal relatives who were becoming too powerful.
Coming back to your Mughal example; the boy-emperor Akbar the Great, once he reached the age of adulthood, ordered his former guardian, and the regent of the state while he was a child, Behram Khan, to go for the Haj (pilgrimmage) to Makkah.
Behram Khan died on the way.
I don’t know if this was a general practice or only enforced by the Mughals, but to banish individuals from the land, they would force them to go to Hajj.