Is the mother entitled to custody of her two children (4.5 years and 1 year 9 months) given her desire to travel with them to Lebanon, where the security situation for Syrians is unstable, keeping in mind that the father resides in Saudi Arabia and the mother is Syrian?

Most scholars are of the opinion that the father has a greater right to custody when the parents separate, in contrast to some who believe that the mother has a greater right if it serves the child's best interest.

According to all opinions, it is stipulated for the child's relocation that the path and the destination country be safe; otherwise, the parent who remains is more entitled to custody. Based on this, the wife does not have the right to travel with the children if the situation is as you described.