Rationales of transmigration:

Samskaras (inclinations, special abilities) are a proof that an Atman transmigrates.  If a newborn child is born without any trace of Samskara, then it would be almost impossible to teach anything to that child because very probably it would have no groove to receive and/or process the learning.  Since there is Samskara carried on from previous life or lives, the child is able to learn.

Some children are born with crippling diseases, some are born carrying a disabled body and also mind.  How can such painful conditions be explained?  Unless we accept the theory of Karma, such odd conditions cannot be explained.  Some are born is royal families and enjoy all material luxuries apparently without doing any hard work, whereas some are born out of poor parents and suffer immensely.  

It is observed that some very small children show special and extraordinary abilities without having been trained into the same.  Prodigal children provide an telling evidence that in their past lives they have received special orientations and the same is in full blossom during the present life.  

The case of Abhimanyu:  Subhadra, the Yadava princess, wife of Arjuna was carrying and Arjuna was describing the nitty gritty of Cakravyuha, a complicated array of soldiers to trap  an enemy soldier or soldiers, to her with a view that the baby in the womb would know the details. Arjuna narrated the formation and then as he was explaining the method to escape from being killed in Cakravyuha, Draupadi fell asleep. Abhimanyu, the baby in the womb knew how to enter in Cakravyuha, but unfortunately, he did not learn how to escape and got killed in the battlefield.

This story from the Mahabharata tells how Samskara can be carefully imbibed even much before a human being is born.    

Karma: The cycle of ‘Karma, Karma-phala and Samskara (conditioning)’ and the bondage that it creates, are in continuum. The cycle of Karma makes it possible to come to life again and again. It is possible to break this cycle of life and death provided one attains Moksha by completely cleaning up the Samskaras.

Even though one achieves Moksha, once one cycle of creation of Brahma comes to an end, even a liberated soul has to come back to life. This is the logical process how the creation is made to continue forever.

The supreme creative force recycles everything, be it Atman or matter. The notion of Mukti or eternal rest has limited meaning and in right context.  What to speak of human beings, even Brahma cannot have luxury of eternal rest (Mukti).  Brahma has to work day and night without a holiday. So also, the destructive (Mahesvara) and preserving (Vishnu) forces.  They are always on their toes.

It is also substantiated by time (Kala). Time is neither linear nor cyclical. Time has no beginning and no end.

Shri Krishna confirms about both these: a) time being Anadi and Ananta and, b) transmigration of Jiva when he says (Gita.4.5):

Bahuni me vyetitani  janmani tava carjuna/

Tanyaham veda sarvani na tvam vettha parantapa//

“Both of us have passed through many lives.  I know about all of those lives while you may not, O Dear Arjuna!”

We all are here for billions of years and shall be here for all the time to come.

Next comes the process of Transmigration

Transmigration of Atman

            A lucid elaboration about transmigration of Atman appears in the Brihadarnyaka-Upanishad (4.iv.i onwards).