PETROPAVLIVKA, Ukraine—Even before Malaysia Airlines Flight 17 was shot down, the war raging between Ukraine and pro-Russia rebels created crushing challenges for the mayor of this small, worn-down village.

Natalya Voloshina couldn't pay municipal salaries, pensions or energy bills because money from the central government in Kiev was frozen. The coal mine where her husband and high-school sweetheart works largely shut down. The fighting was creeping closer.

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