What is the ruling on dealing with a brother who apostatized from Islam and married a Christian woman and had children with her, and how can the parents' grief over him be alleviated?

Grieving over a brother's apostasy is natural, for the calamity in religious matters is immense. One must differentiate between natural affection and religious affection. Natural affection for relatives, even disbelievers among them, persists, and it differs from religious affection, which is not permissible for a polytheist. An example of this is the Prophet's (peace and blessings be upon him) love for his uncle Abu Talib. You must continue trying to advise your apostate brother in the best possible manner, treating him kindly, and praying for his guidance.