What an interesting question!
The Midrash tells us that the staff was passed down from generation to generation and was in the possession of the Judean kings until the First Temple was destroyed. In Samuel I, 17:40, it is said concerning King David: "And he took his staff in his hand." According to Midrashic tradition, this is a reference to Moses' special staff.
It is unknown what became of the staff after the Temple was destroyed and the Jews were exiled from their land.
Rabbi Menachem Posner