Blusafe Origin is battery operated and works via Bluetooth, therefore you are never locked out. You will still be able to use your fingerprint, key-card, App or wireless keypad if there is a power cut.
The only functionality that will be affected are those that require an internet connection, such as remotely controlling the lock and any Smart Home functions.
Blusafe Breeze is battery operated and has no connection to the internet, therefore the lock will work regardless. If the batteries die, it still has a manual backup key.
Blusafe Mirage is powered by an internal, rechargeable battery, therefore the lock will work providing it has charge. If the battery dies, it can be powered via a micro-usb cable.