Everett Ferguson, in an article, “Origen and the election of bishops” makes the case that in the third century (the fourth doesn’t differ that much), “different modes of selection to church office [were] actually practiced in different regions” (p27). Those modes include
election by acclamation of the whole church community (which is what Cyprian, in the mid 3rd century seems to think is normal)
through election by the presbyters of the city
selection by the predecessor bishop that is then ratified by the people, and ordained by the neighbouring bishops (a practice widespread in the fourth century),
To this, in the fourth century, we can add two more I would think:
Decision by a synod. At a number of the church councils, decisions were made to depose certain bishops from certain sees, and to install other bishops in their place.
Decision by Emperor. Technically Emperors didn’t ‘elect’ bishops. But they exerted inescapable influence on those that held certain sees, most of all Constantinople. Gregory of Nazianzus brief time there is probably due to Theodosius.
Of these options, 3 appears to be the default option where ‘possible’. Elections could be contested, as in the case of Basil of Caesarea’s election. Athanasius election to Alexandria appears to be on the basis of his predecessor, but it was contentious.
A couple of sources to get started with for further reading.
Leemans, Van Nuffelen, Keough, and Nicolaye (Eds), Episocopal elections in Late Antiquity
Everett Ferguson, "Origen and the Election of Bishops," Church History 43.1 (1974): 26-33