I am not sure that "soul" is very well defined across cultures. But it seems to me that, as James Kingsbery observes, interpretations of Christianity could fit your query, as might modern folk psychology.
The soul in much Christian and Neoplatonist doctrine is "simple substance" in that it is continuous and indivisible. Yet outside of Calvinist predetermination is it clearly "mutable," in that it registers the imprint of our actions. And in some Augustinian interpretations, I believe, the defiled soul is not so much eternally damned, as simply lost to divine life, thus in a sense rendered finally "mortal."
In Plato's "Myth of Er" at the end of the Republic, not only are "souls" mutable, they are recycled through a kind of plastics factory of mortal destinies. But I would hesitate to say that souls for the Greeks are eternal. As far as I know, that is not entirely clear and consistent. They did not seem to hold ideas of "linear infinity" in our mathematical or Christian sense. It is more as if "soul stuff" gets melted down into the blanks of entirely new persons.
Meanwhile, many people in our modern secular age believe death ends it all, yet do adhere to belief in a psyche, an object of "psychology," that is treated as a mutable but continuous identity or "simple" substance, even as its parts are regularly replaced. I'm not sure how else one would define a "mortal" or finite "soul."
Finally, there are many instances in myth in which a "soul" is actively rendered finite, as it is "released" from the confines of its perpetuation, as with the Buddha, Dracula, and various other unearthly Wanderers.