I can answer in the case of one patriarch of Constantinople: Nestorius. Firstly, how was Nestorius chosen? His election came in the midst of a controversy over whom to elect to the position and his election was that of an outside candidate, so as not to give in to one of two factions in the city. He was at the time a monk at the monastery of Euprepius, trained in the Antiochene tradition. He was selected by the Emperor and installed in 428 as the Patriarch of Constantinople. To let you know just how much power this wielded, at least within the city walls, within five days of Nestorius' election he sent a police force to an Arian chapel (which was illicit, but there were places in the Empire where the laws were pretty lax) to destroy it. The Arians decided not to go down without a fight, so they set their own chapel ablaze! The police were not ready for this and the fire ended up burning down several buildings in that quarter of the city.

This sort of civic privilege continues in the Empire through the early and late Byzantine periods.

Louis Duchesne, Early History of the Christian Church From Its Foundation to the End of the Fifth Century. Vol. 3, The Fifth Century. trans. Claude Jenkins. New York: Longmans, Green and Co. 1924: 219.

Socrates, Ecclesiastical History 7.32.