Atman is infinite, all pervading, non dual itis, the one reality in all things. When physical body dies, but subtle body does not die, and causal body does notdie. That's what transmigrates, goes to other worlds and other lives. This is called a jiva, sentient being. And this is the one that travels. It's just like the screen of the movie when we find the hero going from one country to another. The screen enables this movement to happen. Atman never comesor never goes, because it's infinite,. All pervading,non dual itis; there are no two realities.
Swami Sarvapriyananda teaches the Katha Upanishad, which is one of the most popular Upanishads. The scripture concerns the story of a young man who goes to the house of death to find out what happens after death. The reply describes the nature of reality and the meaning of life.