Why did Mary and Jesus’ brothers go looking for him and why did Jesus turn them away?

Q: Why did Mary and Jesus’ brothers go looking for him and why did Jesus turn them away? (Matthew 12:46-50)

In Matthew 12:46-50, while Jesus is preaching, his mother and brothers come looking for him. When someone points them out to Jesus, he answers, “’Who is my mother? Who are my brothers?’ And stretching out his hand towards his disciples, he said, ‘Here are my mother and my brothers. For whoever does the will of my heavenly Father is my brother, and sister, and mother.’” (Matthew 12:48-50). (Note that the word used for “brother” in the original text is the word that is also used for cousins, half-siblings, and other kin. See the New American Bible explanatory notes for Mark 6:3. )
Why did Mary and the others come looking for Jesus? We know that they come “wishing to speak to him” (verse 46), but that’s all we know. And though it’s easy to read Jesus’ words as a rebuke of his family, in fact his response is not a sign of disrespect. It’s a way to let everyone in the crowd know that even though they have no blood relations to Jesus, they too can become close to him. True family ties come not through shared blood, but through shared values and desire for God.
With regards to Mary specifically, the National Conference of Catholic Bishops make a beautiful point in their pastoral letter Behold Your Mother: Woman of Faith. “In the preaching of the Savior, His ‘mother’ is whoever hears God’s Word and keeps it. All who truly follow Christ become ‘mothers’ of Christ, for by their faith they bring him to birth in others.”