Three of my four arrived under their own steam and chose to stay. Mutual adoption. They wanted to stay and we invited them in. The pit bull was in the worst shape, having been hit by a car then chased off by various neighbors. After a pat on the head, she crossed the driveway and curled up in a corner and slept all day. I checked on her, gave her a pan of water, and by that evening she decided "this is the place." I was planting flowers when she ambled over, nudged my shoulder and wagged "what are you doing?" That was it--she was in. Animals may not have a way to keep score, and who knows how they keep track of such things, but when you suddenly make their life easier, remove the rough edges, offer food and attention, they are grateful. I see it in this dog every time she heads into her hay-filled dog house in the garage when there's a thunderstorm. Every time another meal is offered with a pat of affection.
Our fourth one was adopted from the Humane Society, and as much as she likes the other animals, she'd be happiest if they all went away and she could just bask in my shadow 24-7.