When people learn that Sophie and I volunteer at a "Kill Shelter," we get a variety of responses. One of the most frequent seems to be a variation on, "Oh, I don't know how you can do that - I couldn't handle knowing that the animals might get put to sleep."
Shelter Diaries' posts would indicate. Given that part of working with a shy, frightened, or headstrong animal is getting to know how it thinks... well, it seems like I wind up making a lot of friends I know I'll lose in one way or another. It's inevitable. The only way to avoid the painful part is not to do it at all. A lot of people take that route, and I can't fault them for it.
It's impossible to "make yourself" stop caring. Empathy has no on/off switch. So you mourn each one that doesn't make it. As Sophie pointed out to me when Buster died, though: "The heart is one of the largest, strongest muscles in your body. Like other muscles, it hurts when you overwork it. And like other muscles, the pain only means it will be stronger the next time you have to use it." Every animal I've worked with has left me a stronger person, including and especially those I've cared about enough to leave me gutted when they die. That strength is something I can bring to the next dog who needs a hand, who I may or may not see leave the shelter.
In the end, you can't focus on the big picture; it'll crush you and drive you out of your head. Each day, each animal, each moment... that's where you have to work. It isn't about winning the war. It's about giving the fight everything you've got. That's what matters.