“Today I ask how your governments will respond to this international crisis. We ask the international community to help us, not because we want to outstretch a begging bowl, but because forces far beyond our control have pushed us to this dire situation,” Mr. Williams told the United Nations on Monday. “Rebuilding Barbuda is not a task we can undertake alone.”

Roosevelt Skerrit, the prime minister of Dominica, where Hurricane Maria made landfall late Monday as a Category 5 hurricane, pressed “friendly nations and organizations” to provide a helicopter so that he could survey the “widespread devastation,” which he described as “mind-boggling.”

In a special session convened by Secretary General António Guterres before the official opening of the 72nd United Nations General Assembly, those Caribbean leaders and others appealed to the body to rethink humanitarian aid. They asserted that because climate change is fueling more intense storms, vulnerable countries must have a better way to recover than to beg for money with each new devastation.

Climate change, they said, is no longer a distant threat. Islands are already suffering millions of dollars in losses that they can barely afford because of planet-warming greenhouse gas emissions baked into the atmosphere, the leaders said.

“Climate change and its consequences should not be a subject of speculation or debate,” Mr. Medina said. “It’s a truth which hits us and which causes great uncertainty.”

Leaders did not make explicit demands at the formal United Nations session. Behind the scenes, though, several said it was past time for the creation of a special funding mechanism to help countries deal with the unavoidable consequences of climate change. No amount of planning in Barbuda, for example, could have protected the island from the utter collapse of its infrastructure, Walton Alfonso Webson, Antigua and Barbuda’s ambassador to the United Nations, said in an interview.