What are the parameters for a woman to object to a non-mahram man, and what are the stories that indicate a woman's accountability towards a non-mahram man?

It is permissible for a woman to enjoin good and forbid evil upon a non-mahram man, provided it is within the bounds of Islamic law. Among the stories of women enjoining good upon men are: Khawla bint Tha'labah's objection to and advice for Umar ibn Al-Khattab, a woman's objection to him when he intended to limit dowries, Aisha's disapproval of some Companions' actions, and other famous da'wah (preaching) stances taken by several female Companions and others.