ZAPORIZHZHIA, Ukraine—Scores of civilians trapped for weeks in the rubble of Mariupol and other towns occupied by Russian forces reached Ukrainian-controlled territory after traumatic journeys as Russia redoubled efforts to seize the sprawling steel plant that had served dozens of them as a refuge.

Among the flow of people arriving Tuesday at a processing center in Zaporizhzhia, a Ukrainian-held city in the country’s southeast, was a convoy of around 150 civilians who had spent weeks sheltering in the Azovstal steel plant where Ukrainian soldiers continue to hold out.