Anyone can prove to themselves, using scientific methods, that they perpetually remain the same conscious entity while the body perpetually changes. It’s simply a matter of hypothesis and observation. Depending on what kind of science you accept as valid, the Bhagavad-gita gives a thoroughly scientific explanation of transmigration of consciousness through different bodies, especially in the eighth chapter. You may also want to examine Dr. Ian Stevenson‘s three thousand case studies of children remembering past lives.
A conscious being, atma, may conceivably be transferred to any part of the material world in the next life. One’s destination after death is based on the decision of God and His representatives. Our karma and our desire is factored in.
The soul leaves the body at death and does not remain in the body.