What do the Vedas say about child custody?

Interesting question.

Firstly the Vedas contain no laws whatsoever. Laws which are loosely and remotely based on Vedic principles are found in the Dharma Shastra and there is very little about child custody.

Divorce was not a common practice and so the issue of child-custody after a divorce was never mentioned. Legal texts usually only address the common issues of the day.

There are many laws regard adoption – who and how they can be adopted.

The common unit was the joint family set-up which means in the case of a death or disappearance of a parent then the child is raised and cared for by the joint family with no specific person appointed as a guardian.

I do not know what the current laws regarding the matter are in India