Jacob A Lollar has submitted to Academia, "The Meaning of Ephesians: Competing Christianities in Second Century Ephesus". We all know "Ephesians" had nothing to do with Ephesus. Lollar has taken upon himself to explain what "Ephesians" was trying to do; and, why it got associated with Asia's capital city later.
Lollar notes that Ephesian Christendom was better associated with John. For some values of "John" that is even true. The Revelation came from a nearby island, and Ephesus was first in the Apocalypse's recipients. There exists also that answer to Luke's Acts, John's Acts. (The various versions of Luke/Acts floating about, don't matter.)
Given the above, I do wonder how come Ignatius hasn't been noted. To Ignatius is ascribed another letter to Ephesus. Unlike the Pauline "Ephesians", nobody serious contests Ignatius' content and recipient.
Ignatius parallels some Pauline phrases. Clearly he meant to evoke 1 Corinthians. Colossians and even "Ephesians" 5:2 also get parallels. To dovetail Lollar I do not read where Ignatius says "as Paul wrote to you". Compare how the Romans under Saint Clement levied Paul's Corinthian correspondence, also against the Corinthians. (At least 1 Cor, also I think the Lachrymose Letter which we don't got no more.) Rather, Ignatius is accepting "Ephesians'" consensus of Asia Minor Pauliana. To the extent "Ephesians" possibly exists to smooth over dispute over Colossians, that dispute was no longer a problem in Ephesus, anyway.
Ignatius faced a different problem in Ephesus, shared with those who accepted that collection like Ephesus' bishop, Onesimus. The problem too was shared, as Ignatius writes to bolster the bishop. The dispute was getting to the point of violence. Ignatius argues that as they are cruel, do not become cruel in return. (Ignatius pretends the cruelty is all done by the baddies, as usual in disputes. We may ignore this.)
I don't think anyone involved knew the Lukan corpus; Ignatius' appeals to the life of Christ are famously paracanonical. Luke's absence means Acts of John weren't there either. Ignatius further parallels some events now found in our Gospel of John, like the anointing as integral to the Passion. But Jesus' very title was "the anointed". Some anointing scene should be reified in any useful narrative. In fact such a scene is not restricted to John 12. The parallel to John 12:32 is bereft of Johannine tropes so likely precedes both.
As to what the schismatics actually taught, Ignatius brings the Physician Creed, which has become famous: There is one Physician who is possessed both of flesh and spirit; both made and not made; God existing in flesh; true life in death; both of Mary and of God; first passible and then impassible — even Jesus Christ our Lord.
Ignatius implies that the other team by not accepting this cannot be cured. One might read this creed to bar the doors of episcopal hospitals. I do not read such: instead, Onesimus and Ignatius were dealing with Christians who refuse medical treatment on grounds flesh be nothing compared to spirit. Christian Scientists, one might say.
The Physician Creed is at the core of Ignatius' thought generally. Ignatius must elsewhere end-run around Mark 15 as he cites Peter against Christ being an asomatic spirit. Some skeptics even wonder, as a result of Ignatius sidestepping Mark, if Mark's Gospel yet existed. I think this Gospel did exist by then, but Ignatius can't use it. (Mark elsewhere is facing headwinds, as was the Gospel in the Egerton papyrus. We are lucky to retain Mark.)
I suspect that Onesimus' problem was the spiritual tradition, beyond even Mark. This tradition can be identified with that around the Acts of John, which - again - wasn't yet written, but a lot of similar lore had been written. The tradition may already have been associated with John. Ignatius won't dignify it with John's name, but that doesn't mean his enemies hadn't. Ignatius' angle was instead to bang the codices of Paul.