Across the English Channel, the U.K.'s white cliffs beckon. On fine days, men and women with children in their arms and determination in their eyes can see the shoreline of what they believe will be a promised land as they attempt the perilous crossing clandestinely, ditching belongings to squeeze aboard flimsy inflatable boats that set to sea from northern France.

In a flash, on one recent crossing attempt, French police swooped in with knives, wading into the water and slashing the boat’s thin rubber — literally deflating the migrants’ hopes and dreams.

Some of the men put up dispirited resistance, trying to position themselves — in vain — between the boat and the officers’ blades. One splashed water at them, another hurled a shoe. Cries of “No! No!” rang out. A woman wailed.