So does the person chosen to be the avatar change? Not in my story.
Does the appearance of the avatar change? While it stays very similar, it can change.
Are avatars reborn? Yes, just like the gods, they are eternal.
Does this mean avatars can die? The godflesh can die.
Is there any proof of this? You can look for photo similarities throughout history to real people but the embodiment of a god is not meant to be known by humans, let alone proven to them.
In my stories, there is also the case of mortals trying to take the place of gods. There are also the cases of demi-gods (hybrids) and Eve’s (and sometimes Adan’s) continual obsession with being the king/queen of everything and the god/goddess of all. This often ended up with confusing and overlapping mythologies.