A family of five is seeking asylum in New Zealand because, they say, climate change is making life too dangerous in their low-lying island homeland in the Kiribati islands. They are going to court later this month to argue their case as climate refugees.

New Zealand has twice refused to let the family stay because they aren’t political refugees, the usual reason people seek asylum in other countries.

This is the beginning of what could be wholesale flight from places slowly being inundated by rising seas linked to climate change. A recent U.N. report makes clear that coastal areas around the planet—from the Arctic to the South Seas—are feeling the impacts; and while many people are making elaborate plans to live with higher water levels, others like this family are planning to flee.

A story about the family’s plight in The Guardian offers scant hope for the family’s chances:

Legal experts consider the … case a long shot, but it will nevertheless be closely watched, and might have implications for tens of millions of residents in low-lying islands around the world. Kiribati, an impoverished string of 33 coral atolls about halfway between Hawaii and Australia, is home to about 103,000 people and has been identified by scientists as among the nations most vulnerable to climate change.

The man (family father) said that in about 1998, king tides began regularly breaching the sea walls around his village, which was overcrowded and had no sewerage system. He said the fouled drinking water would make people vomit, and that there was no higher ground that would allow villagers to escape the knee-deep water.

Returning to the island would endanger the lives of his two youngest children, he said.