Is it a condition for the mahram to remain with the woman in the country she traveled to, or is his presence necessary only for her travel en route?

A woman should not travel except with a mahram. If her mahram travels with her and she finds safe accommodation, he is not obliged to stay with her. The mahramiyah (presence of a mahram) is a condition for travel only. It is also stipulated that the woman should not be in seclusion with a non-mahram man and that the place be safe.