Finite things are in constant change, that is part of the definitive characteristics/qualities of finite things.
Of course, everything physically existing in our spacetime is constantly changing. But I have no difficulty conceiving of a universe where this rule does not hold. Constant physical change may be a property of objects in this universe, but I don't think that it is a logically required property of finitude. Consider the resurrection body. It is a physical resurrection, because we are physical beings, and our bodies are part of our identity. It is a perfected body, in that it will not age or suffer illness, and it is free of other restrictions we have in this world. But it is still physical. Presumably, this body will be unchanging. Our minds, I would suggest, will be capable of change and growth--we could continue to learn. But our bodies will remain unchanging (unless, perhaps, we choose of our own volition to change them).
Our concept of "self" does change as we age. But this changing self is connected by the chain of memory. I propose a threefold definition of what it means to be a "self":
1. A single, autonomous, conscious being, distinct from others. (i.e., no "group mind" or consciousness).
2. An awareness of ourselves as being what is described in #1.
3. Memory of what we have experienced.
If, in the afterlife, I were to lose all memory of what I experienced on Earth, then I would no longer be "me". The same is true if I lost my distinctness or my awareness of my distinctness as an individual being.