The early church did indeed apply this language to specific cases, particularly in relation to God's judgment and church discipline.
For example, Chrysostom describes Judas and Job as being "delivered to Satan," and says:
Many such instances still occur. For since the Priests cannot know who are sinners, and unworthy partakers of the holy Mysteries, God often in this way delivers them to Satan. For when diseases, and attacks, and sorrows, and calamities, and the like occur, it is on this account that they are inflicted. (Homilies on 1 Timothy, V)
Jerome doesn't seem to disagree with Lucifer of Cagliari's description of the priest's typical practice:
The priest, we know, offers his oblation for the layman, lays his hand upon him when submissive, invokes the return of the Holy Spirit, and thus, after inviting the prayers of the people, reconciles to the altar him who had been delivered to Satan for the destruction of the flesh that the spirit might he saved ("The Dialogoue Against the Luciferians," 5)
And Basil of Caesarea offers specific instructions regarding those who have repented from grievous sin:
There can, however, be no doubt that we ought to receive those who have repented of impurity committed in ignorance for thirty years. In this case there is ground for forgiveness in ignorance, in the spontaneity of confession, and the long extent of time. Perhaps they have been delivered to Satan for a whole age of man that they may learn not to behave unseemly; wherefore order them to be received without delay, specially if they shed tears to move your mercy, and shew a manner of living worthy of compassion. (Letter 188, VII)
The question of what the church fathers meant by the phrase is a bit different, but the context of the above quotes should be helpful in that regard.