What should I do, Islamically, regarding my mother and sister traveling to Western countries without a mahram, given that my mother broke her promise and went without my knowledge? Is it obligatory for me to disavow them because of the Prophet’s (peace be upon him) saying: "I am innocent of every Muslim who resides among polytheists"? To what extent is a Muslim obligated to forbid those under his care from evil, and is it obligatory to abandon them if they insist on disobedience, and how should this abandonment be?

Your duty is to advise your mother and sister against the wrongdoings they commit, while bearing in mind that the mother's right is immense. Therefore, it is impermissible to sever ties with her or treat her poorly, no matter how much she deviates from the Sharia. Rather, advise her with gentleness and respect. It is incumbent upon a man to remove evil from those under his care, unless doing so would lead to a greater harm. As for shunning a disobedient relative (other than parents), it depends on what serves the greater good; if shunning deters them, then it is permissible, but if maintaining ties is more beneficial, then that is preferable.