Must a mother offer expiation for an oath (kaffarat yamin) because of her vow to help a poor family for life, knowing that the family members are no longer poor and there are other families more in need?

The mother must fulfill her vow according to her intention. If she intended to help a specific family for the rest of their lives, she is not permitted to break the vow or divert it to others. However, if she did not have a specific intention, and the reason for the vow was the poverty and need of that family, then she may stop the vow to them or divert it to others if that reason ceases to exist. This is because a vow is like an oath, and its expiation is the expiation of an oath.