Consider the equivalent case today: heredity.
When you are born, your parents pass on their skills to you. They also pass on other things; their network of contacts, their money, and so on.
If you inherited the mind of someone powerful, you would inherit the location of the money that they cached for themselves towards the end of their previous life, in the assumption that they would be coming back for it - a mnemonic for a swiss bank account or whatever.
You would inherit the contacts, the other powerful people - friendships in your previous life would continue. So would enmities.
You are essentially immortal; almost any power you might have gained from living forever, can be a power you gain from being reincarnated with perfect recall.
It will become a commonplace in wills that one's inheritance would pass not to one's descendants, but to one's reincarnated self. That business empire would be placed into the hands of a pro-tem manager, but when you come back and key in the code to identify yourself, you take over the reins again.
Legally, it would take very little time before ownership and identity were redefined to account for these temporary lapses of existence. Some debts would last past death; you couldn't get an education, then kill yourself to avoid the student loans but keep the knowledge.
The death penalty would not be given.
War would be very different. The purpose of genocide would be to skew the demographics of one's foe, to scatter them over the earth, and to separate them from their property long enough for it to be seized.
Murder would be considered less serious, though probably still seen as the most serious assault you could perform on someone.
Suicide would go down in some groups (no longer an "easy way out" for the depressed), but up in others (it becomes a way of rebelling with only relatively minor consequences; there would be groups of people who enjoyed childhood enough to want to remain children indefinitely, killing themselves when they get too old).
Birth rates would go down, WAY down, to the point that population growth may reverse. Having a child that's the creation of the two partners, and helping them form their personality, and make their start in the world is one thing; but the risk of playing mommy and daddy to a cuckoo who was created and mentally formed by an earlier set of parents hundreds or thousands of years ago... not appealing to most. We're selfish about kids. There's a reason adoption and fostering levels are always lower than they need to be.
As birth rates fall, the probability of getting a reincarnated person becomes a certainty.
Passing on our body's genes is as close as we get to immortality, but in a system of guaranteed reincarnation, bodies become irrelevant. We don't need to pass on genes.
Instead, we pass on our selves.
Another point of kids is care into our old age; that would no longer be an issue. If you become too old to look after yourself, then kill yourself, and you'll immediately get loving parents who will look after you.
The only people who'd have kids would be those who actively enjoyed child-rearing; or those who were paid to have them by some social schemes paid into by the dying (a "birth tax" charged on the dying?)
So... drifted off-topic a little, but yeah - the social connections you made, and the financial and business inheritances you had, would give an enormous first-mover advantage over those who weren't reincarnated; but there would be fewer and fewer births, so reincarnations wouldn't be guaranteed.