What is the ruling of Islamic law regarding demanding the apprehension of adulterous women and their children, and how did the Prophet, may Allah’s prayers and peace be upon him, deal with them if they were not to be stoned?

If "encompassing" means "care," then the child of adultery must be cared for and provided for by his mother. If she fails to do so, it becomes incumbent upon the Muslim community to undertake it. As for the care of the adulterous mother, it is Islamically required, unless it is in a way that assists her in adultery or obstructs a punishment ordained by Allah. The Hadith of Al-Ghamidiyah indicates the ruler's concern for the adulteress and her child. It is also obligatory for anyone capable to deter the adulteress from adultery to protect society.