Does a divorce take place if a man says to his wife: "If any of your family visits you, or you visit them, or you meet them, then you are divorced," in response to an action by her brother, and is there an expiation or a way out of this oath?

It is forbidden for a Muslim to insult or demean his brother. Your wife’s brother’s action is forbidden, and he must repent and seek reconciliation. As for you, you should pardon and forgive. Do not burden your wife with her brother's bad manners by preventing her from visiting her family.

Regarding the conditional oath of divorce: divorce occurs, according to the majority of scholars, when the condition is met. If it was the first or second divorce, you can take her back; otherwise, she is not lawful for you until after she marries another man. However, Ibn Taymiyyah holds that if the intention was merely to threaten and not to divorce, then only an expiation for breaking an oath is due. If you were not aware of what you were saying due to extreme anger, then divorce does not occur. It is advisable to consult the competent Sharia authorities specializing in personal status matters.