So in my experience thus far, The Matrix probably has the clearest illustration of a quadra progression-regression cycle: Alpha->Beta->Gamma->Delta->Gamma->Beta->Alpha.
(Spoiling Warning! People who haven't seen all of the movies probably won't understand what's going on here, and it will ruin the story, so turn back now.)
Here is the timeline I am using for this analysis:
The Matrix timeline
The story starts, obviously, before the movie begins. I would consider the initial Alpha phase to be the invention of machines that culminated in them being able to think for themselves; it is characteristic of the end of an Alpha stage to let things get away from them and mess with the sorts of things that they don't have the capacity to deal with. They are essentially overlooking Se: not respecting the destructive power of that with which they are dealing (other stories like Frankenstein, Pandora's Box, and the scene from Fantasia where Mickey creates a bunch of broom drones that almost drown he and his master come to mind; the end of Alpha represents an archetypal loss of innocence of sorts, brought about by creating something that can't be controlled). Riots break out in the human world after a machine acts on its advanced impulses and kills it's masters.
The first Beta sequence of the story begins when the divide between two organized factions (the Aristocratic "Us vs Them,"), the humans and the machines, is created. The machines separate into their own city, Zero One, and begin making plans to make a stand against humanity, In the end, they wind up taking a classic Beta kamikaze-esque end-game risk, attempting to destroy their enemy in one fell swoop at extreme cost to themselves, by "scorching the sky," blocking out the sun, the source of the machines' energy. Unfortunately this measure fails to stop the machines, and humans lose the war.
The machines, having gained dominion over the humans, enter the Gamma phase by "stopping the buck," so to speak, of human destructive advancement, and attempting to cope with the extreme circumstances that have been created as a result of the humans' kamikaze attack. The machines realize that the humans themselves can be used as resources, machines in their own right, for bio-electric energy that will power their civilization. They construct a functional society based on this new technology and, after a few errors, develop a system with which they can control the minds of humans effectively.
The first Delta phase, representing stability in society and a lack of change due to functional sufficiency, begins after the invention of the effective form of The Matrix. For 5 cycles, 500 years, human minds are held in captivity by the Matrix and the cycles of Zion's planned destruction, and the machines rule the world without threat.
The return to Gamma from Delta consists of a similar anomalous occurrence that causes Alpha to progress to Beta: the emergence of forces that cannot be controlled by existing systems. However, instead of causing two groups to band together and fight against each other, this transition signals the breakup of established alliances, and leads up to the eventual reorganization based on newly vested interests.
Everything in Neo's time is going as planned by the machines, until the program called Smith declines to return to the source upon failing to fulfill its obligations to the system; after having his prime manifestation within the Matrix destroyed by Neo, Smith, the most powerful program within the Matrix construct, goes rogue and begins to replicate himself in an attempt to spread his own influence: systematic destruction of humanity and the Matrix that supports its life. In addition, Neo breaks the cycle planned by the machines of, sacrificing the collective interests of humans for the sake of saving the one he loves, Trinity.
The second Beta stage consists of the solidification of the new groups whose forces are aligned against each other: Neo against the Smiths, and Zion against the Machines. While Commander Locke and the forces of Zion do battle against the onslaught of the machine army, Neo takes the drastic action characteristic of the Se/Ni half of the Beta quadra by venturing to the machine world unarmed in an attempt to make peace with the machines. He battles Smith, and, emerging victorious, reconciles the conflict between the humans and machines by saving the machine world from the Smith virus. The humans and the machines accept their respective places in the world they have created by their conflict.
The final Alpha phase is the time of co-habitation between humans and machines, the "new solution," wherein both attempt to rebuild their civilizations independent of one another. All humans are unplugged from the matrix, and it is assumed that they begin reconstructing their respective societies on the grounds of maintaining peace with one another.
The Matrix acts as an effective metaphor for Quadra Progression not only in its representation of the various stages, but also in what these clearly illustrated stages allow us to observe about the dynamics of each stage, and what they say about each function.
The clearest divide is between times of peace and conflict: the Se/Ni stages are characterized by clear divides in power, whether it be between specific groups (Beta) or divided between individuals (Gamma), whereas the Si/Ne stages are typified by either a power monopoly that enforces peace (Delta) or cooperation that necessitates peace (Alpha).
The rational divide is slightly more subtle, in that it represents the duality of innate human instinctual traits (Ti/Fe) and progressive organizational/functional traits (Te/Fi), but in the Matrix series it is clearly characterized by whether greater power is wielded by the humans (Ti/Fe) or the machines (Te/Fi).