The term 'eonic' was probably a poor shoice of terms: it merely reflects a discrete sequence in a continuous stream, e.g. a clock, a discrete number of hours in continuous time.

Another term is 'punctuated equilibrium': the eonic sequence resembles a series of punctuations (the default state is not really in equilibrium) in a frequency pattern. The term is intuitive, but we use it to start, and then change terminology: it is somewhat else's term and has been grafted onto Darwinism, spoiling its use.

We will speak of transitions and a frequency hypothesis: the transitions come about about every 2400 years, visible most unexpectedly since the invention of writing ca. five thousand years ago. The transitions show massive social innovations and cultural transformation in a limited period, with results diffusing into their environment. The transitions also can occur in parallel, e.g. the Axial Age. The modern transition from 1500 to 1800 shows a massive cascade of innovations, and concludes around 1800. The modern era is thus shot out of canon and diffuses rapidly to create for the first time a global civilization.
We see that 'eonic evolution' deals in general culture, religion, philosophy, science, politics, art/drama/literature, etc...

We can guess with some confidence that this sequence goes back into the Neolithic and no doubt beyond, but it would seem that the 'evolution' of homo sapiens concludes its sequence of transitions with the modern transition. Was active during the emergence of homo sapiens, and then shut down at the start of the Paleolithic as man, perhaps in Africa, evolved in a distinct transition sequence and zone, and then began to disperse globally sometime around a hundred to fifty thousand years ago.
The model must make a distinction what the transitions do and what its free agents make of it. This means that 'evolution' in our sense must be able to act directly on the human mind either as a group or an individual. The idea of a creative state might help here, although such concepts remain uncertain in this context.

Let us note one key fact: the 'eonic evolution' of civilization is real evolution but has no genetic component: it is a pure form factor process. That suggests the same is true of organismic evolution: there is some evolutionary process beyond the genetic process, that latter confusing biologists ad infinitum.

The eonic evolution process must connect with earlier organismic evolution: perhaps the connection lies in the idea of the evolution of freedom. The animal emerges as a locomotive creature whose life/action is a primordial history, and the progression of animal forms from 'free locomotion' to advancing primates to hominids with increasing powers of will, intention, and freedom, as free action in various modes. Thus animal evolution passes into the human evolution of groups inside species, culture that is, and the seed source of civilizations springing out the 'group' aspect of evolution inside a species.

This makes good sense of the total mystery of human evolution but it is obviously very incomplete. And cannot pinpoint the 'mechanism' but descriptively suggests that tandem evolution of individual and group is the core process.
This process is teleological, but we must be wary of the concept. The distinction of system and free agency means that the system can direct evolution only in transitions leaving free agents for the remaining interval. The idea that Man Makes Himself remains crucial, but with the obvious catch.

This discussion implies design, but that has no theological implications. The design in nature is not a proof of any kind of divinity. Further, we cannot even describe man and his potential, or his consciousness and 'will' But progress could be made on those areas. Our discussion should be considered naturalistic, but clearly we hardly understand nature.

I am a de facto 'atheist' in quotation marks but that applies only to the 'god' of classic monotheism. I cannot reject in advance possible future concepts of cosmological scope, with the 'inside' and 'outside' issue possibly disappearing into the jaws of Kant's antinomies.
The reason for the previous statement is that we confront the remarkable history of the Israelites who noticed the eonic effect in their history, recorded a transition filled with prophets whose visions produced the first stage of new literature in a religion. But we can see now that the 'god' idea doesn't work, but that there was an obvious confusion between an emergent religion of 'god' and the wrong explanation of the eonic effect as theological. Note that nearby in India, we see the emergence in parallel in a related transition, of the atheist religion of Buddhism (emerging in relation to an older Jainism). The god concept will confuse out discussion, and the Israelites sensed this and were actually reluctant to use any such term as 'god', viz IHVH as pointer or glyph! Their insight rapidly degenerated into the pop cults of 'god' that emerge later.

The point here is that we are closer to Bergson's 'creative evolution', and idea I can't endorse except to wonder if the concept of creative energy in relation to a teleological process can be made to make sense.
Lamarck had the gist before Darwin muddled everything: his one liner is something like, 'evolution' shows a rise in complexity on one level and an interaction with an environment on another, the distinction of macro and microevolution. This distinction once lost with Darwinism turned 'evolution' into a hopeless muddle.