Is it permissible to authorize one's family in the home country to pay the expiation for an oath (kaffarat al-yamin), or is fasting obligatory when one is unable to feed the poor in the country of travel?

Your fasting is not sufficient as long as you are able to delegate one of your family members to purchase food or clothing for ten needy people in your country. Financial expiations permit delegation, even if the one obligated is capable of performing them himself, because what is obligatory in them is the giving of money, and this is achieved through the action of the delegate, as mentioned in the Kuwaiti Encyclopedia of Islamic Jurisprudence and by Sheikh Ibn Uthaymeen.