What is the ruling on a father remaining in a non-Muslim country due to the custody of his son, and is he entitled to take his son, who is living in a non-Muslim environment, to a Muslim country?

We pray to Allah to help you remain steadfast. If the mother is a disbeliever or an apostate, she has no right to custody of her Muslim child. If she is a Muslim but is immoral, or if there is a fear that she will raise the children in a deviant manner, then the father has the right to demand custody. The divorced mother forfeits her right to custody if she marries someone other than the child's father. If it is possible to bring the woman to a Sharia court or to retrieve the child by any other means, then do so. If that is not possible, and if you fear for your faith, then flee with your faith and supplicate for her guidance until he reaches the age when custody ends, then take him and teach him the correct methodology. However, if you do not fear for your faith by residing in a land of disbelief, and you are able to call him to Islam and preserve his faith, then that is better than your traveling and leaving him.