SLOVYANSK, Ukraine—Ukrainian authorities are scrambling to evacuate the remaining civilians from the eastern Donetsk and Luhansk regions as Russia begins its new military offensive here and pitched battles get closer to the area’s main population centers.

The main cities in the Ukrainian-controlled parts of Donetsk and Luhansk, such as Kramatorsk, Slovyansk and Severodonetsk, have already turned into ghost towns, with almost all stores and businesses closed, streets emptied and only a handful of apartments in each housing block still inhabited.

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