Transmigration, the term used by Jews, or Reincarntion, used in eastern religions, was a concept of Judiasm based on Torah scripture. My guess is that they got it from the Greeks. It is clear that the concept of the soul came from the Greeks. The Greeks were big on the soul and heaven, and reincarnation was considered a reasonable possibility, but not really an article of belief. Same for the jews.

The early christians inherited the concept of transmigration from Jewish thought. After all early christianity was just another form of Judiasm among many.

However, when Constantine made Christianity into a state religion, transmigration was abandoned. I believe that it was abandoned because having only one chance to make heaven allowed for much better control of Romans subjects. If you just get reborn and get another chance, then it is not so important to do exactly what the Church and the State wants. This working model persisted all through the dark ages and is often thought to be responsible for the dark ages.

The dogma about 1st & 2nd birth, etc., was never stated by Jesus. If we all followed what Jesus wanted, we would all be practicing Judiasm today. The dogma was just the means of the church and the state to control people. Some of us have broken away from that control by reading the teachings of Jesus and ignoring the dogma. But then again some of us have not.