Is it permissible for a custodial mother to travel with her children (3 and 6 years old) without their father's permission, whether she is still his wife or divorced, a distance of 600 km to her village where the marriage contract was performed, and would such travel nullify her custody, transferring it to the father?

The mother has a greater right to the custody of her young, undiscerning child, unless she remarries or there is another impediment. If she is still married to the child's father, she must reside in his home. If not, the majority of jurists stipulate that her custody of her child must be in his father's country. If it is impossible for both parents to reside in one country, the majority of jurists are of the opinion that the mother's right to custody is forfeited, and custody transfers to the father, provided that the father does not intend to harm the child. However, if his travel is for a temporary need, the child remains with whichever parent is resident. Some jurists, including the Hanafis, permit the mother to move with the child to a nearby country where the father can see him, or to her original country if certain conditions are met. The fundamental principle in custody is the child's legitimate best interest. Therefore, custody may remain with the mother even if the father is the one who moves, if the child's best interest lies in remaining with his mother.