The headlines are misleading, but the truth behind them is strange and elusive, from a Western perspective. A "right to custody" has appeared in headlines on CNN, in Khaleej Times, and in news links circulated on social media. That wording, at least the way it would commonly be understood in the U.S., is completely wrong.
Just as divorced or separated parents in the U.S. do, Muslim Saudi women who get divorced, or whose children later reach the age for living with their fathers, have the right to ask a court to decide who gets custody, and to have the court consider the case.
What is new this month is apparently a procedural reform: IF the parents have no disputes on child-related issues, the mother can get custody by filing an application with the court, instead of going through a full-scale court case. The Justice Minister's circular says, in part:
A mother may submit a probate application to the competent court for certifying her custody of her children, provided she signs an acknowledgement that there were no existing disputes ...
For granting custody to a mother, the judicial panel considers her capacity for custody and then determines her application in accordance with Sharia and legal requirements, without the need for initiating a lawsuit, as is the case with all probate certifications indicated in Chapter 13 of the Law of Civil Procedure.
-- quoted in "Saudi mothers can now retain custody of children without filing lawsuits" by Habib Toumi in Gulf News
Almost all the news stories include that key phrase, "provided there are no disputes," but the headlines and lead sentences, and indeed the rest of the wording of each article, totally ignore it, as if it were a technicality or an unthinkably rare and meaningless exception. This is as bad as the reporting on no-fault divorce laws or covenant marriage laws -- blowing up changes to sound far more drastic than they are, by making crucial exceptions sound like meaningless recitations, and naively ignoring or belittling the role of agreements and disagreements between divorcing spouses.
There is no change in favor of foreign or non-Muslim women, as far as I can tell.
Other substantive changes the Minister announced:
The circular also gives the mother the right to carry out all formalities related to her children at government departments, embassies, education offices and schools, and to apply for and collect her children’s passports.
She will also be able to collect all child support and maintenance from government and civil entities, but may not travel with her children outside the Kingdom without a judge’s permission.
-- "Divorced Saudi mothers win new rights to child custody" by in Arab News
To see what this is a change from, here is what looks like the most up-to-date background on child custody in Saudi courts:
"THEMATIC REPORT ON MUSLIM FAMILY LAW AND MUSLIM WOMEN’S RIGHTS IN SAUDI ARABIA," report to CEDAW, February 2018, by Musawah: For Equality in the Family