The idea of an evolution of human consciousness throughout history has been a part of spiritual thinking for more than a century - I know it mainly through considering the work of Rudolf Steiner, Owen Barfield and William Arkle over the past couple of years.
(I encountered the idea over thirty years ago summarized in the work of Colin Wilson, but did not then pay much attention.)
The idea of an historical evolution of consciousness seems to go-with a belief in reincarnation, because reincarnation allows each person to participate in the different stages of evolution that are aiming-at a fully divine form of consciousness.
Steiner and Barfield describe this aimed-at state in some detail - in essence it combines on the one hand a direct involvement with, and participation in, reality such as was characteristic of early man and remains characteristic of early childhood; with, on the other hand, a fully alert, self-aware, purposive and analytic consciousness which is characteristic of the adult consciousness and the modern phase of Western history.
So, the idea is that I am personally experiencing the distinctive modern, alienated consciousness now - including the knowledge and aspiration towards a future state; however, in earlier lives I have also personally experienced, and benefited from, earlier phases of human consciousness. At some point later this life, and perhaps further lives, I may incrementally, a step at a time, learn how to combine the positive qualities of all phases. This aimed-at fully divine conscious state is what Barfield calls Final Participation.
According to Steiner and Barfield, these earlier life phases include non-incarnated lives - lives when we were conscious but had no body. So the theory is really one of multiple lives, rather than re incarnation.
Therefore the human spirit or soul (i.e. that entity which is reincarnated) is here conceptualized as undergoing an educational process toward which each life is contributing.
Repeated lives, many lives, seem to be necessary in order to allow for the very large amount of experience and learning required to bridge the gap between being a man and becoming a god. Certainly, one mortal life seems grossly inadequate for this, especially given that most human lives in history were terminated either in the womb or in early infancy - a small minority of humans have reached adulthood, and even fewer of these have had a full experience of marriage, family, maturity and growing old etc.
So, evolution of consciousness and reincarnation seem to make a neat package. However, this package is, if not incompatible with Christianity, at least somewhat alien to the structure of Christianity; which places a great deal of emphasis on the individual life which we are experiencing now, and sees 'this life' as having potentially decisive consequences for eternity.
And certainly, while reincarnation seems to described in the Bible - most notably in the case of John the Baptist apparently being a reincarnated Prophet Elijah - there isn't any scriptural description of a scheme of reincarnation as the norm. And especially not of multiple lives.
My interpretation is that ancient Christianity saw reincarnation as true, but as an exceptional possibility, done in exceptional cases and for specific purposes - rather than as the standard procedure for the majority of people.
Does an exclusion of reincarnation then rule-out the evolution of consciousness throughout human history? No, but denial of reincarnation with multiple lives does limit the role of evolution of consciousness in the lives of individual spirits or souls - it breaks the link between the evolution of consciousness in history and the evolution of my consciousness and the specific consciousnesses of every other individual.
Put differently, the arguments which (in particular) Owen Barfield makes for different types of consciousness in human history, such as his insights into the changing scope and meaning of words, may well be true; but they lose their relevance to the evolution of my consciousness and your consciousness if we were not present (in earlier lives) actually to experience the several stages of this historical evolution.
In sum, the historical evolution of consciousness is a matter of historical but not personal interest, if we ourselves were not present during that history.
My own belief is therefore that I accept Barfield's description of human consciousness having changed throughout history and in broadly the way he describes; and I also accept that we are meant (or destined) to achieve that mode of consciousness Barfield terms 'Final Participation'. But I do not accept that the two are causally linked - for instance I do not believe that I have, myself, personally participated in the historical phases of the evolution of consciousness during previous lives.
Rather, I see the evolution of consciousness as a sequence which is recapitulated in different scales in different situations: e.g. through human history, in each person's individual development from childhood to maturity, and also in the largest cosmic scale of our salvation and divination across eternity.
I therefore would modify the Steiner/ Barfield model, since I regard this evolutionary sequence of consciousness as a basic and necessary process in terms of Man as a whole and also individual men working towards fuller divinity. And I think it is because the process is basic and necessary that we see it appearing and re-appearing here and there throughout reality; operating at many scales and across many time-frames.