You can activate “Trojan Blast” when a face-down monster shifts, such as with “Creature Swap”. But no damage will be inflicted because it was face-down.
You cannot activate “Trojan Blast” when a monster shifts control to become an Equip Card. So you cannot activate “Trojan Blast” when “Relinquished” or “Allure Queen” equips itself with your monster.
“Trojan Blast” is not a targeting effect. Select the monster when “Trojan Blast” resolves.