Shifting performance is unchanged regardless of remaining battery charge.
The components do provide indicator lights for battery life. See this article to know when to charge your batteries: How do I know when to charge the batteries in my eTap derailleurs or replace the shifter batteries?
The derailleur remains in the gear it was already in when the battery runs out of charge. Batteries are interchangeable between front and rear derailleurs to help get you home if you don’t have a spare battery.