A person is a process

There is a famous tale of Indo-Greek king Menander meeting the Buddhist philosopher Nagasena as recorded in Buddhist canon Milindapañha. In it, Nagasena explains to the visiting king the Buddhist philosophy of Shunyata. That there is no soul (or Atman) and hence no life after death. To illustrate his point, he uses the analogy of a chariot. If you separate the chariot into its constituent parts like the wheels, the carriage, the horses and reins etc. then each part in itself is not the chariot. But only when put them together in a particular configuration, we give it a name of chariot, for our convenience. When a chariot is dis-assembled into its parts, the chariot's soul does not escape anywhere because there was no soul to begin with.   

It is a powerful argument.

I think, the modern science adds a bit more to this argument. As we understand it today, a living thing is a process. Cells in our bodies are constantly dying and new cells are being generated continuously. The food we eat and air we breathe provide the energy and raw materials for this regeneration.

So, to add to the Buddhist argument, living things are of course a sum of constituent parts and additionally those parts are continuously being regenerated. Living being are a process.

A better analogy that the chariot would be comparing living things to a fountain. A fountain is a process not a thing. When it stops, the process ends. The fountain soul does not go anywhere.