September 24, 2015 12:37 PM EDT

From the end of August through mid-September, as he followed the migrant trail from Turkey through Greece, the Balkans and Hungary, what struck Yuri Kozyrev the most about the thousands of asylum seekers he saw through the lens of his camera was their resilience in the face of demeaning hardship. They never seemed to give in to despair. They tried hard to look dignified – mending and washing their clothes, shaving and tending to make-up—even as they walked for days on end through blistering heat on their way to Western Europe.

“You might expect to see these people broken by this point,” Kozyrev, a contract photographer for TIME, told me as we watched hundreds of them pass in an unending stream across the border from Serbia to Hungary—and into the European Union—after traveling for more than a thousand miles by land and sea to get there. “But they are smiling,” he said. “They are happy.”

It is in some ways a testament to the hopelessness they left behind in the war zones of Syria, Iraq or Afghanistan that they endure the long journey to Europe with such hopeful looks on their faces. Even as they confronted the razor-wire fences and the detention camps where they are forced to wait for registration, they held their heads high.

In his photographs, Kozyrev doesn’t just capture the moments of terror and exhaustion that the migrants endure – although these are, of course, abundant along the road they take. He also shows the poignant displays of integrity and self-respect, as when they pray at a mosque they built out of plastic netting or stop to preen in front of a traffic mirror on a highway. The resulting picture is not of shattered lives but ones in the midst of a grueling transition, looking ahead to the process of laying down roots in a place where they can feel safe. “That’s what drives them,” says Kozyrev. “Hope.”

Yuri Kozyrev is a TIME contract photographer represented by Noor.

Alice Gabriner and Mikko Takkunen, who edited this photo essay, are TIME’s International Photo Editor and TIME.com Associate Photo Editor, respectively.

Simon Shuster is a correspondent for TIME magazine.

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