The general consensus is somewhat varied because of Russian and the so-called Abkhazian "country's" attempts to fabricate history of their own, but most accept that "ethnic" Abkhazians are Georgianized north Caucasians with the "Georgianization" starting after the spread of Christianity. Most of them were fully Georgian and term "Abkhazians" was used to describe Georgian people from that area until 15-16 century when it became extremely hard to understand just who was who.

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The main problem that almost all people seem to face is to discern difference between the people I just mentioned and the modern "Abkhazians" that live the region, so a little history about it: Starting from around 15-16 century the whole region became increasingly difficult to govern for Western Georgian kings and dukes as Ottoman turks pushed into black sea and Islam started spreading in Northern Caucasus. With the area heavily devastated from Pirates, raids from muslim and sometimes even pagan mountain people most of the people moved into lowlands and further south. This is a somewhat well documented fact and I can provide Russian and European sources confirming it. Anyhow - with Georgians moving southwards and abandoning their old living places, the mountain people started migrating downwards, abandoning their tribal way and starting to have contact with both Georgians and Turks. The ottomans were more successful, as they managed to spread Islam heavily, while Georgians were busy with infighting and could only defend themselves as the newcomers started raiding them just like others

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The name of these tribes are several, they were sometimes a federation of many smaller ones, but two of them are accepted as the "main" ones - Apsuas and Abazas. These people, starting with the retreat of Georgian population slowly, but steadily moved south and started living in the deserted areas left over by Georgians. As Georgia was fragmented into several parts it was not able to cope with a full blown migration and the best that was done was the construction of a "great wall" by Levan II Dadiani that attempted to wall off Samegrelo (Mingrelia) and parts of the Abkhazia from those tribes. As those tribals started living in Abkhazia permanently, they started calling themselves Abkhazians - the name that stuck and was adopted by Georgians as well. And while their migration was successful, parts of the Abkhazia would still be Georgian (Especially the north-western and eastern part) and it was not until the advance of Russian empire until the area was fully conquered and duchy of Abkhazia fully abolished in 1864, with the last duke of Abkhazia Mikheil (Michael) Shervashidze being arrested and forcefully relocated in Russia

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I'm sure the post is already big enough so to sum it up - the region of Abkhazia was inhabited by north-western Caucasians that were Georgianized starting from 5-6 century, later the area was almost abandoned and was settled by another north-western Caucasian people once more that adopted the old name and started calling themselves Abkhazians. If you'll have some questions feel free to ask, this is a very hard topic for most of the people to have some info as most of the legit information is available either in Russian or in Georgian only from those times

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If there are any typos sorry about that and also sorry for writing Abkhazia/Abkhazians too many times lol