Both of these stories by Flannery O'Connor are social critiques set in a changing American South, and both feature an elderly female protagonist who is in some way struggling with social change.
In "A Good Man is Hard to Find," a prominent theme is religious hypocrisy. The grandmother likes to think of herself as the guardian of the old, superior ways, and she doesn't much like the sassy behavior of her grandchildren or newer ways of parenting. This is seen in how she treats her son. She thinks nothing of making demands on him that inconvenience the rest of the family with regard to their shared vacation destination. She holds on to racist attitudes and uses epithets that are offensive and decidedly not Christian. Even so, when she attempts to talk the Misfit into sparing her life, she tells him, “If you would pray, ... Jesus would help you.” There is no evidence to suggest that she turns to God other than at times of need, such as when the murder of her family and herself is at hand.
Julian's mother in "Everything That Rises Must Converge" is also struggling with changing times: her racial attitudes are being challenged in public and within her own family. She was raised in a prominent white family in the South and can't cope with the idea that African Americans are on the rise and will not stand for her belief that they are an inferior race that must remain under white control. Her son Julian challenges her throughout the story, and he is mortified and angry when he tells her, “There are no more slaves,” and she replies, “They were better off when they were."