The gospels of Matthew and Luke are edited versions of the gospel of Mark.
It has been said that the four gospels are different eyewitness accounts, which contain factual inconsistencies but confirm the basic story. This explanation, however, cannot explain why these two later gospels have birth stories and the gospel of Mark does not. Nor would eyewitness accounts be likely to have instances of precisely the same wording. This kind of evidence implies that the authors of the gospels of Matthew and Luke had a written copy of the gospel of Mark when they wrote their own gospel accounts. Both added birth stories, and also made other changes to the gospel of Mark in order to present their own versions of the church's witness to the good news in Christ.
Might the differences and similarities of the first three gospels be explained by changes over time in a single oral tradition? As Paul's letters are the earliest extant written documents of the church, we would expect evidence of an oral tradition to show up there. The following statement by Paul in his first letter to the church at Corinth seems to be such a tradition: "For I handed on to you as of first importance what I in turn had received; that Christ died for our sins in accordance with the scriptures, and that he was buried, and that he was raised on the third day in accordance with the scriptures, and that he appeared to Cephas [Peter], then to the twelve. Then he appeared to more than five hundred brothers and sisters at one time, most of whom are still alive, though some have died. Then he appeared to James, then to all the apostles." (1 Cor. 15:3-7)
This is likely an oral tradition that Paul received from the apostles in Jerusalem, and it is the heart of the gospel stories written later and attributed to Mark, Matthew and Luke. Yet, Paul's brief summary of this core teaching is not at all like the gospel stories. It says nothing about the life and teachings of Jesus, his last week in Jerusalem, and the reasons for his arrest and crucifixion. Moreover, Paul affirms a resurrection appearance to James, the brother of Jesus, which is not recorded in any of the New Testament gospels.
Does it make sense to suppose that this summary of Christian faith was expanded into four oral traditions, which were then written down as the four New Testament gospels? Or, is it more likely that this and other fragments of tradition, perhaps both oral and written, were used by the author of the gospel of Mark to create a story that was later edited, using additional materials, by the authors of the gospels of Matthew and Luke?
Either way, of course, we must assume that Christians elaborated the story. Whether intentionally or inadvertently, whether in oral or written form, the summary of events in Paul evolved into four more detailed stories, which have many similarities but also differ in interpretations. An example may make this clearer. In the gospel of Mark Jesus begins his ministry with a call to repentance and faith because the kingdom of God is at hand. (Mk. 1:14-15) In the gospel of Matthew Jesus begins his ministry with the statement, "Repent, for the kingdom of heaven has come near." (Mt. 4:17) In the gospel of Matthew Jesus speaks almost always of the "kingdom of heaven" rather than the kingdom of God, and unlike the gospel of Mark he launches his ministry without an explicit call to "faith."
In the gospel of Luke Jesus begins his ministry without speaking either of repentance or faith, but by reading in the synagogue in Nazareth from the prophet Isaiah: "The Spirit of the Lord is upon me, because he has anointed me to bring good news to the poor. He has sent me to proclaim release to the captives and recovery of sight to the blind, to let the oppressed go free, to proclaim the year of the Lords' favor." (Lk. 4:18-19, quoting Is. 61:1-2)
Are these different memories that were passed down by oral tradition? Are they the same memory, changed by years of oral tradition? Or, are they, at least in part and significantly, choices made by the authors of these three gospels?
Those who claim each gospel is the literal, infallible or inerrant word of God reject the argument that the authors of the gospels of Matthew and Luke edited the gospel of Mark in writing their own gospel accounts, because this understanding acknowledges that the gospels as compositions. However, it seems clear to most readers of the New Testament that the authors of the gospels of Matthew and Luke, using oral or written materials or both, have edited the gospel of Mark in the way each of them thought best to proclaim the good news of the gospel in Jesus Christ.
Were they inspired by God? I am sure they thought they were, and I believe each was. But the gospels we have are clearly not evidence of a "word for word" transmission from God. The authors of these gospels were inspired by their faith. They wrote different gospels because they had different materials, as well as the gospel of Mark and a "common sayings tradition." But they also understood the Septuagint in terms of the experience of their different churches and drew on its wealth of teachings and images to highlight diverse aspects of God's character and purpose.
Of course, the gospels of Matthew and Mark join the gospel of Mark (and Paul) in proclaiming that the death and resurrection of Jesus are good news for all those who repent and have faith. Despite their differences, they express this core teaching of the early churches. Moreover, they contain a collection of common sayings attributed to Jesus that are not in the gospel of Mark. These are not only similar in written form, but are woven by the authors of the two later gospels into the gospel story of Mark in much the same order. That implies that this "common sayings tradition" was not simply a collection of oral traditions, but existed as a written text when the gospels of Matthew and Luke were written.