This effect can be activated at any time in the Damage Step, even During Damage Calculation.
Sending "Blackwing – Kalut the Moon Shadow" to the GY is a cost to activate its effect.
The effect of "Blackwing – Kalut the Moon Shadow" does not target.
You can send "Blackwing – Kalut the Moon Shadow" to the GY during either player's turn, as it is a Quick Effect.