I was reading a book by the late Louis Bouyer entitled the eucharist and I found it fascinating although a little too deep for me.
In it he covers the liturgical development of all of the major rites. He stated that the liturgy generally was structured from the Jewish Berekoth prayers as used in the Synagog plus the Berekoth prayers used in the home which directly form the eucharistic prayers.
The interesting thing to me about the Roman liturgy is that he maintains that it has successfully retained some discernable elements from the first two centuries in spite of all of the development in the intervening years. And do you know what liturgy he said was most like the early Roman liturgy? The early liturgy of Saint Mark from Alexandria!
It is not that one derived from the other, but that they had parallel development under similar influences. Certainly churchmen travelled from one community to another and they had an influence knitting the practices together. This was easy because they were all using Greek then, even in Rome!
Bouyer stated that there were two “families” of liturgies (at a minimum) at the outset within the Roman Empire proper, the Antiochian-based group and the Roman-Alexandrian axis. The primary influences that made the difference were what kind of Jews predominated in a locality. Syria and places north had some very old diaspora communities while Alexandria and Rome had that plus a great many conservative Palestinian Jews (presumably newer arrivals). The structure of how and in what order the prayers were said in the Synagogs influenced what order they were said in the Christian worship. These people were from Synagog communities and many continued to attend both until they were excommunicated by the Synagogs.
This is important because from the beginning people just did what they knew. When the first generation was passing away and conversions from Judaism slowed to a trickle, greater care was taken to train new presbyters, and the liturgies began to take on a rigidity that we think of as a ‘style’ or ‘tradition’.
Separately and not in the book: there is also the tradition that Saint Mark travelled with Peter, and was sent by Peter from Rome to establish the community in Alexandria. Both theories could in fact be true, or either one. In any case, the two liturgies were similar in the early years.
The early Ambrosian rite was similar to or derived from the Antiochian tradition, then it fell under the influence of the Roman rite.
Anyway, I know that there is a great deal of speculation in this telling, but I think a lot of this info has merit. I have focused on the first 300-400 years specifically, the subsequent years are much better documented and much has already been said.