The world of Bleak House is so interconnected that duty is the natural byproduct of almost any relationship. Because no characters are wasted or non-recurring, and anyone we meet we will meet over and over again, the people in the novel have no choice but to form ties of obligation and reciprocity with everyone they meet. In a closed system like this one, duty creates and maintains reputation; ensures continuity between friends, neighbors, and acquaintances; and creates links between people whose lives would otherwise be so divergent that they would never intersect.
The main duty of women in Bleak House is restoring, keeping up, and increasing connections between characters. This can be for the good: Esther takes it as a given that part of her life will involve helping Caddy and Miss Flite, while Mrs. Bagnet thinks nothing of going, without being asked, to find George's mother. It can also be negative: Mrs. Snagsby does her best to connect Mr. Snagsby and Jo not just as patron and charity case, but as father and son, while Hortense maliciously tries to bring together Esther and Lady Dedlock. Without this female duty, the world of the novel would collapse, or stop moving forward.
In a world as oppressive as this one, characters who manage to completely evade responsibility and duty are actually impressive and worthy of admiration.