For purposes of this question, reincarnation works in the following manner:
The transmissible personality, memories skills, knowledge and experiences of a person shall be referred to in this question as a Soul.
During life, the soul develops alongside the body. At death (the moment that the soul's body ceases to be a suitable vessel to hold it, some time after the cessation of all processes of life), the soul becomes separate from the body.
Some time after the death of its previous body, the soul may attach to a newborn member of the same species. As the newborn matures, the soul progressively integrates with the body and becomes fully integrated at physical maturity. The period of time between death and reincarnation may be as little as a second, or may be many lifetimes, though it is most commonly a relatively short period of time. It may be speculated that one or more 'bad' lives may delay reincarnation, however good lives may also delay reincarnation. There is some correlation between the reincarnation times of a particular soul. Reincarnation is not guaranteed.
While reincarnation often occurs in a region in which the reincarnated individual lived, it need not always do so, and could theoretically occur at interstellar distances (with lightspeed delays). Reincarnation may occur into a newborn of any race or gender of its species provided that the newborn has a reasonably normal brain.
A reincarnated individual gains the mental skills, knowledge and memories of the soul which attaches to it.
a newborn without a reincarnated soul is not born with a soul of its own, it develops one at some point during adolescence.
The effects of brain injury and/or illnesses such as dementia do not become part of the soul. A soul may be considered to function additively, and has no mechanism for externally applied subtractive processes.
A soul's content degrades slightly between reincarnations. A soul which has reincarnated many times may typically remember roughly ten previous lifetimes, more if the previous lives were short, or less if the previous lives were long. More recent previous lives are more clearly remembered than older lives.
Rarely (on the order of 1 in 100,000 reincarnations), a soul may reincarnate into two newborns simultaneously (taking lightspeed delays into consideration). Such duplicate reincarnations lead to two separate souls, the souls do not become one again after the death of both of the bodies.
Where the species' population numbers are rising, there will be many newborns who are not born with a reincarnated soul. Where the species' population numbers are constant or falling, more newborns will have reincarnated souls, to the point where a newborn without a reincarnated soul is very rare.
Obviously there would be profound social implications for this sort of reincarnation, but they are not the point of this question. However, the social implications will have to be addressed if they could affect the evolution of the species.
My question is, What effect would reincarnation have on a species?
As an example, humans do not demonstrably, frequently and reliably reincarnate. However, for purposes of this question, let us suppose that at the time that hominids speciated from chimpanzees and bonobos, roughly 5.5 million years ago, the hominids gained the ability to reincarnate. If all other selection pressures remained effectively identical and evolution progressed similarly wherever reincarnation has no evolutionary effect, what differences could we expect between real-life humans and reincarnating humans after a similar period of time since speciation 5.5mya?
These differences need not be purely physical, and may be differences in the basic psychology of the species.
There seems to be some confusion as to how souls work. They do not function as a metaphysical cloud storage that's accessible at any time, like a google drive account that's able to have its content downloaded into RAM relatively quickly.
Rather, a soul's content must be copied into the physiological memory of the body during maturation before it becomes accessible, and any content that is in excess to that which the body can retain is culled according to the usual physiological processes that discard or overwrite less useful memories.
After the body's brain is physiologically mature, the soul isn't able to add much more of its content to it, perhaps only 5% of that gained during maturation.
The process by which memories of former lives may be lost is that memories successfully written into the body are removed from the metaphysical soul. During life, a new soul is written from the body's brain. When the body dies, the remainder of the old soul is copied in a slightly lossy manner to the new soul before the new soul goes on to be reincarnated... when and if it does so at all.
Perhaps 1 in 20 souls do not reincarnate at all, and perhaps 1 in 6 do not reincarnate 'immediately' (within a single typical lifespan of a member of the species).