They die.
They leave their sack of meat behind.
They then wander the Earth seeking purpose in their afterlife. But there truly is no greater purpose for them or special meaning to their afterlife, they're just like dust in the wind.
...Which is why there could be a character who uses ghosts and makes pod people/moving manikins out of them. Ghosts are a resource to them. This character could incorporate this into their moveset (you're making a fighting game IIRC, right?). Maybe make this character Hades or some similar character? Maybe even the Grim Reaper? Whatever works best for the mythological themes of the game I guess.
There could also be a character that acts as a rival/foil to them. Maybe a Buddhist monk type of character. Since Buddhism is all about the purification of one's spirit, this character, upon discovering the hapless spirits wandering without purpose (maybe he's the only human who can actually see them, or he's part of a temple of monks that is specifically trained to "hunt" lost spirits), decides to set up some kind of sanctuary for spirits, where they come to go about their afterlife in peace, as if being part of a community of the living once again. Since the word "Buddhism" is derived from the word "budhi" (which IIRC is part of the sacred Pali language), which translates as "to awaken" in English, I think this could be an appropriate way of mixing the Ancient Greek interpretation of spirits with the Buddhist interpretation of spirituality.
I don't know how well something like that fits into the context of your game but it's something to work off of, I guess. Buddhism may not strictly fall under the typical definition of "mythology", but all classic mythological universes of lore were once strictly followed as religions. Both mythology and religion are quite malleable for the context of creative works, as they really represent a people's culture more than anything.