While it isn't the best analogy, I see some similarities to how Jesus went to seek and save the lost. As incredible and pronounced as the change is in animals that are rescued and re-homed, it pales in comparison to the transformation faith in Jesus brings to a person inside and out. Jesus did not lure people with food or secretly set a trap to capture prospective followers for their own good: He simply sought them out, spoke the truth with love and compassion, and He was the Servant of all. Over time people listened to Jesus and responded to His call to follow Him. Belief was only the beginning of a process of being transformed from within by the power of the Holy Spirit.
Jesus had compassion on the multitude and also to a sick, lonely man who did not have anyone to help him. Jesus knew what people had suffered at the hands of others, due to their own sin, choices and folly and chose to reveal Himself to them as the Son of God through Whom forgiveness of sins and eternal life was assured. While some scampered away, others drew near and sat at His feet. Jesus demonstrated patience continually with everyone around Him, whether they were snarling enemies or fawning disciples. This patient compassion seen in Christ ought to mark Christians as well, for though a person can be born again in an instant there is a process of maturation and wisdom that cannot be rushed. God sanctifies us by faith in Christ, and we are also learning to sanctify ourselves by walking by faith and not by sight.
The patience shown with neglected and sick animals can put me to shame because I have not always extended such patient compassion to other people. It is easier to be compassionate towards a stray than your own dog who should know better from much training. Better than seeing ourselves as the one out there looking for strays in the analogy, it is humbling and helpful to know we are one of those strays. We are not a Saviour; we have no power to transform anyone. But we can lead those in our pack to seek the LORD Jesus Christ Who loves and has compassion on us, provides for us, protects us, our Master we need right now more than ever. Praise the LORD for Jesus, the One who seeks and saves the lost.