Perhaps you want to put a dollar amount on it. If the bill reaches $X, they have your permission to let the dog go. Or maybe you want the caretaker to make the decision after receiving input from a veterinarian you know and trust. Maybe it’s about hospitalization. Your dog has kidney disease but has been able to remain at home, yet when the only way to keep him alive is at the clinic, it’s time to put him down.
The more specific you can be, the better the chance your dog has of being treated according to your wishes should you be unavailable to make decisions yourself. “Here’s his food; I’ll see you a week from Thursday” is not enough for a dog getting on in years.
The people who care for your older dog in your absence are not the only ones in his environment warranting special consideration. Newcomers to your household will affect your friend as well.