Three ways of reincarnation are known. First, when the soul moves from one person to another. This is تناسخ) tanasukh). Some people believe they had an earlier life and are convinced that they ever existed in the body of another person. The second, when the soul is transferred from man to animal تفاسخ) tafasukh). There are beliefs that the good and bad deeds committed in human life are rewarded by transferring the soul either to a better or worse quality after death. Thus, as a punishment, someone in the next life may become an animal, a pig, a cockroach. But there is also the assumption that the soul moves into inanimate things and objects تراسخ) tarasukh). Some African animist beliefs profess that their souls can move into trees, phenomena. These statements are logically inconsistent with the revelation of the Quran, the Day of Judgment, and accountability:

Every man's fate We have fastened on his own neck: On the Day of Judgment We shall bring out for him a scroll, which he will see spread open. (Quran 17:13)

Fate: Tair, literally a bird, hence an omen, an evil omen, fate.36:19. The Arabs, like the ancient Romans, sought to read the mysteries of human fate from the flight of birds. And many of us in our own day seek to read our future fortunes by similar superstitions. We read in the previous verse that there are Signs of Allah, but they are not meant to subserve the vulgar purpose of disclosing our future destiny in a worldly sense. They are meant for quite other purposes, as we have explained.
Our real fate does not depend upon birds or omens or stars. It depends on our deeds; good or evil, and they hang round our necks.
These deeds, good or evil, will be embodied in a scroll which will be quite open to us in the light of the Day of Judgment.

A similar thought is found in 84:6-12. A fate fastened to the neck is not a pre-written fate that one carries and is written in advance. This would be a predestination that makes the judgement on responsibility arising from free will impossible. It's just the opposite. The aya speaks about the fate that man went through and deeds committed according to his/her own intentions and that all is recorded. One of the fundamental tenets of the doctrines of Islam is the free will and its judgment in the Afterlife. The doctrine of Youm al-Qiyama,القيامة يوم the Day of Resurrection is associated with the Last Judgement. The Quran writes about it in many places. The reincarnation is a denial of that doctrine. This particular destiny, which is recorded and takes everyone with him/her to this day, cannot be confused with other destinies. I can only carry my own destiny, nothing else. Not even other person can take my fate with him/her.
This is the essence of logic, but it does not change the fact that those who believe in reincarnation must be respected. As we expect everyone to respect our faith. However, it is necessary to get to know each other's thinking, to discuss similarities and differences. We also need to know which are the cornerstones that cannot be changed and what are the items that should be harmonized in order to follow the needs of the age. Reincarnation is an item that Islam does not and cannot profess.