TALLINN, Estonia—On a recent weekend in an Estonian forest, a group of new recruits in army fatigues practiced maneuvers. A sergeant major barked out instructions and the unit changed formation, sometimes crawling in the dirt between birch trees.

But these weren’t professional soldiers: They were some of the hundreds of edgy Estonians who have flocked to a volunteer army in the months since neighboring Russia annexed part of Ukraine to the south.

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