Is it a condition for one who hastens departure from Mina that the agent completes the the throwing of the pebbles, and is it permissible for a pilgrim to appoint an agent for the throwing of the pebbles while he (the pilgrim) is in Mecca?

A pilgrim capable of performing the rami (stoning) is not permitted to depart from Mina on the first day of nafr (departure) except after performing the stoning of the jamarat. It is permissible to appoint a proxy for the rami for the incapable, the sick, children, and those in similar situations. If someone is permitted to appoint a proxy and does so, they are allowed to depart from Mina before their proxy performs the rami, but they are not permitted to perform the farewell tawaf until their proxy has completed the stoning.