I’m currently reading through the Apostolic Fathers (Michael Holmes’ translation is excellent, I highly recommend it). When I was reading the Didache I came across this passage:
Therefore appoint for yourselves bishops and deacons worthy of the Lord, men who are humble and not avaricious and true and approved, for they too carry out for you the ministry of the prophets and teachers. (15:2)
So, it seems that perhaps the earliest catechal document we have from the early church instructs congregations in a given city to appoint and ordain their own episcopal overseers (who are, of course ostensibly in communion with overseers from in all other local churches). Moreover, if we have have local churches consecrating their own bishops, how does that jive with any straightforward theory of apostolic succession?