The following are quotes taken from Pope Innocent’s responses to the letters sent on behalf of the Council of Carthage and the Council of Milevis, written around 416 and 417 respectively. The numbering is based on Augustine’s letter collection, which includes these two letters. I will highlight portions that seem of particular interest for understanding his role as bishop of Rome, as well as the role of councils and tradition. [Fun fact: Pope Innocent was a native of Albania!]
181.1. In examining questions about God, with which it is proper that bishops and especially an authentic, legitimate, Catholic council deal with the greatest care, you have observed the patterns set by ancient tradition and have been mindful of Church discipline. In that way you have added strength to the vigor of our religion by true reason, no less now in consulting us than earlier when you issued the decree. For you wanted it to be referred to our judgment, knowing what is due to the Apostolic See, since all of us who have held this position desire to follow that apostle from whom the episcopacy itself and the whole authority of this name is derived. Following him, we know how to condemn what is evil and to approve what is praiseworthy, just as we approve the fact that in observing the teachings of our predecessors you did not think that they should be ignored. For they established it not by a human but by a divine decision that one should not regard as settled, whatever questions are dealt with, even in distant and remote provinces, before it comes to the knowledge of this see. In this way a correct declaration is upheld by the whole authority of this see, and–just as all waters go forth from their original source and the pure waters of their incorrupt spring flow through the different regions of the whole world–from this see the other churches learn what they should teach, whom they should absolve, and whom a stream fit for clean bodies should avoid like those persons filthy with a foulness that cannot be purified. [A note: Augustine uses the language of “stream” and “spring” in Letter 177, apparently in the same sense used here.]
181.2. …you are demonstrating your solicitude for the good of all, and you ask that we decree through all the churches of the world what at the same time benefits all.
182.2. You act conscientiously and appropriately, therefore, in consulting the office of the Apostolic See, that mystical office, I say, to which, except for those matters that lie outside, there pertains the solicitude for all the churches over what judgment should be maintained in troubling affairs. You have followed the form of the ancient rule, which you know has been observed with me by the whole world. But I set aside this issue, because I do not believe that this has escaped the attention of Your Wisdom. Why did you endorse this practice by your action if it was not because you knew that responses always flow from the apostolic fountain through all the provinces for those who ask for them? I think that, especially when a question of faith is discussed, all our brothers and fellow bishops ought to refer it only to Peter, that is, to the source of their title and dignity, as Your Charity has now referred this question, which could benefit all the churches in common through the world. For these churches must necessarily become more cautious when they see that the inventors of these evils have been separated from communion with the Church by decrees of our judgments in response to the two synods.
182.6. …[Pelagius and Caelestius] are excommunicated from the Church by the authority of our apostolic power until they come to their senses (2 Tim 2:26).
[The translation cited is the New City Press edition of Augustine’s works.]