How do black dwarfs die?
We do not know.
Black dwarfs are hypothetical as it is estimated that they are formed when a White dwarf cools down and is unable to radiate energy in about a 100 Billion to a trillion years. Since the universe is still young compared to this massive time Scale we do not know if black dwarfs even exist.
Based on what Black Dwarfs are (theoretically speaking), there would be no way for them to "die" as they would already be "dead".