There are two main versions of the mechanics of reincarnation that I will call "old" and "new" for sake of simplicity.
The old system has been around for thousands of years. While the details differ slightly, it is the system of reincarnation taught primarily by Hindus and Buddhists, but also by some ancient Christian sects and in Judaism until about the fifteenth century. A good outline is provided in an essay by Ken Wilber called "Death, Rebirth, and Meditation". There exists some disagreement as to what survives and how long the whole process takes, but in general rebirth takes between 49 days and 200 years; some say personality and memories do not transmigrate and others do. All the differences are really pretty minor when viewed against the similarities regarding how the mechanism itself works. Human psychology is basically the same as what modern science has shown us. Our fears and addictions carry on with us where we have to deal with them in very real terms. Psychological traumas we suffer in this life must be dealt with in this life or the next, and if we cannot overcome our problems than we will be carried aloft by them into a new rebirth. Some say we meet up with people we've known in this life (or even other past lives) in the between life stage, and that certain souls transmigrate together, and others say we do this all alone, but every action in this life always has an equal reaction in the next. There is no escaping causality, the books must be balanced, full stop. We work through our troubles, gradually perfecting ourselves lifetime after lifetime until we finally break free from suffering all together and abide as Spirit forever.
In the new system, promulgated by certain spiritualist circles and "new age" movements, the whole old system is turned on its head. Rather than beginning in ignorance and spending lifetimes trying to end suffering, in the new system we all begin enlightened and we come to this life to experience suffering. Every soul is enlightened, we come to this world with a sort of willful amnesia, and when we die we are automatically restored to perfect knowledge. The lives we live are games we play with one another, and they don't really mean anything on the other side. When we die there is no balancing of good and bad deeds, because there is no good or bad, as we plan our lives out in minute details from the beginning. People who get raped or murdered make agreements with other souls who will be doing the raping and murdering, so there are no villains and victims, and not because the world is ultimately illusory, but because the world is a game we play for life experience. In the new system souls are basically masochists. You can't really know what something is like until you experience it first hand, so a soul living in perfection might have an idea what suffering is, but it doesn't really know until it experiences it for itself.
Why anyone would want to partake of such experiences I cannot understand. If I were Superman and I knew with absolute certainty I could never get hurt I still wouldn't care to know what it felt like to drive a car into a tree or get shot in the face. I wouldn't care to know what it was like to drown or get exploded. I wouldn't care to know what it was like to live in poverty on the streets, or for that matter what it would be like to live in a mansion eating lobster every day.
The new system is very distasteful and reeks of wish fulfillment. Imagine a group of souls floating around planning their next life together. One soul says to the others "I'll be the child and you'll be my parents. At age five, no four and a half, I'll get cancer and die and break your hearts. Then, fifty years of misery later, you'll die and we'll meet up back here and I'll point and laugh and say 'fooled ya!' Then, the next go around I'll be the parent and you two can be the children who die. It will be rousing great fun!" The idea infuriates me, to be honest. I think that's much worse than the materialist idea that when you're dead you're dead and there's no justice. I would much rather live in a universe where life is ultimately meaningless and there is no afterlife than have a universe where the Nazis made an agreement with the millions of people murdered in the Holocaust that the whole thing would be staged just to see what it would be like to gas eleven million people. Admittedly the grotesquery of such a scenario is not evidence that it is not true in itself, but it does not help the case either.
It is also not helped by the types of evidence that are available to support either system. The old system, which recapitulates everything we know from earthly psychology, is attested to through near death experiences that reveal our mental state and personalities are pretty much the same after death as before, and through spontaneously remembered past lives of children that often turn out to be phenomenally accurate when later investigated. Enlightened sages also attest to the veracity of the old system through the remembrance of their past lives.
The primary evidence for the new system is through hypnosis of adults, that sometimes does produce highly accurate results (which could be more akin to a form of remote viewing), but a lot of times these memories are false, crafted by the imagination of the subject and the leading of the hypnotist. The new system is also, sometimes, attested to by channeled material, but by no means all. Some channeled material flatly denies the existence of reincarnation, other spirits admit they don't know, and some affirm the old system. The spirits that talk about group souls, of instant enlightenment following death, of coming to Earth to experience things no sane person would ever want to experience such as disease, disasters, and war, and flat out deny the existence of good and evil, seem to me to be lower spirits. Whether malicious or simply tricksters (like Internet trolls of the spirit world), they will say whatever gets them the most attention, and earthly interlocutors receiving confirmation of their desires pass it along in books that say you can literally create your reality and that morality is for squares. As John admonishes in his first epistle, we are to test the spirits to see if they are of God. Anyone from the other side can say they are advanced spirits with brilliant insight into the workings of the universe, and that is why we must use our reason and intelligence to see if what they say makes sense.
In the light of the evidence and the moral implications I would have to say that if reincarnation does exist it seems far more likely that the old system with a gradual progression from ignorance to perfection is correct.