EDGARTOWN, Mass. — Immigration lawyers rushed onto ferries to reach a quaint red church turned shelter. Students from a high school Spanish class sought to help translate. Impromptu volunteers scoured a dollar store for clothing to donate.

The frantic activity showed just how unprepared the vacation retreat of Martha’s Vineyard was for the arrival on Wednesday of nearly 50 Venezuelan migrants on planes from Texas.

“We had no inkling of what was going on,” Geoffrey Freeman, the director of the Massachusetts island’s tiny airport, said on Thursday.

The decisions by Gov. Ron DeSantis of Florida to send two planeloads of people to Martha’s Vineyard, and by Gov. Greg Abbott of Texas to send two busloads to Vice President Kamala Harris’s residence on Thursday — roughly 150 migrants in all — were their most conspicuous attempts yet to provoke outrage over record arrivals at the border, a circumstance the Republican leaders blame squarely on President Biden.

But while the transfers of migrants were called out by Democrats as political stunts, they also served as vivid reminders of how ill prepared heavily Democratic parts of the Northeast are to handle influxes of poor migrants, even though they have long been immigration strongholds where many people are eager to help.

On Thursday, Mr. DeSantis spoke proudly of chartering private planes to drop mostly Venezuelan migrants on Martha’s Vineyard. He did so several months after Mr. Abbott and Gov. Doug Ducey of Arizona, a fellow Republican, started sending frequent busloads of migrants to Washington and New York, straining resources in those cities. Mayor Muriel Bowser of Washington declared a public emergency last week in response to the arrivals; on Wednesday, Mayor Eric Adams of New York said its shelter system was “nearing its breaking point” after more than 11,000 migrants arrived in the city since May.

“All those people in D.C. and New York were beating their chest when Trump was president, saying they were so proud to be sanctuary jurisdictions,” Mr. DeSantis said in the Florida Panhandle. “The minute even a small fraction of what those border towns deal with every day are brought to their front door, they all go berserk.”

He did not shed light on who rounded up the migrants — who had been processed by federal immigration authorities at the border and released — and got them on the planes, or how. Several of the migrants, who spent the night at a church in Edgartown, Mass., said on Thursday that after being released from Border Patrol custody in Texas, they had been approached by a woman named “Perla” who offered them seats on planes to Massachusetts but did not say they would be landing on a remote island. This year the Florida Legislature set aside $12 million to transport migrants out of the state.

Mr. Abbott took responsibility for directing buses to the Naval Observatory in Washington, which serves as Ms. Harris’s residence. In a statement, he said the more than 100 migrants were from Colombia, Cuba, Guyana, Nicaragua, Panama and Venezuela.

Ms. Harris “claims our border is ‘secure’ & denies the crisis,” Mr. Abbott wrote on Twitter. “We’re sending migrants to her backyard to call on the Biden Administration to do its job & secure the border.”

The two governors, who could someday face off as presidential primary rivals, do not appear to have coordinated their transports, though both were clearly intended to draw the attention they received.

The Texas governor’s office had previously talked with Mr. DeSantis and his aides about supporting Mr. Abbott’s existing effort to bus migrants out of Texas, said Mr. Abbott’s press secretary, Renae Eze. But Mr. DeSantis appears to have opted instead to create his own spectacle. (Ms. Eze said in a statement that the Texas governor’s office was “not involved in these initial planes to Martha’s Vineyard,” though she added that “we appreciate the support in responding to this national crisis and helping Texans.”)

In Washington, a Fox News camera was positioned near the vice president’s residence to film the drop-off, officials said; volunteers eventually took the migrants to a church. Members of the White House were particularly frustrated that Fox News had apparently been alerted, but not the city government or nonprofit organizers waiting for any potential migrants at Union Station.

The Biden administration portrayed Republicans as determined to harness anti-immigrant sentiment ahead of the congressional midterm elections. But it is unclear what action, if any, the White House can take to stop the drop-offs of migrants, many of whom plan to seek asylum in the United States and are guaranteed that right. Once migrants have been released by border officials and served documents to appear in court, they are no longer in federal custody and are free to travel across the country as they please.

In a speech at the Congressional Hispanic Caucus Institute’s annual gala on Thursday night, Mr. Biden lashed out at Republicans.

“Instead of working with us on solutions, Republicans are playing politics with human beings, using them as props,” he said, his voice tinged with anger. “What they’re doing is simply wrong, it’s un-American, it’s reckless.”

He added that there was a process for handling migrants. “Republican officials should not interfere with that process by waging these political stunts,” he said.

Theresa Cardinal Brown, a former policy adviser for Customs and Border Protection, said a separate question to investigate is whether the migrants were encouraged to board the flights or buses with misleading information.

“There’s no immigration consequence for anyone facilitating their onward movements in the United States,” she said.

Since Mr. Biden took office, more than one million migrants have entered the country, many of whom will wait months or years for hearings because of legal and procedural backlogs. More than a million were similarly admitted on a temporary basis over a two-year stretch of the Trump administration, according to data analyzed by the Migration Policy Institute; these migrants are distinct from the many who enter the country undetected.

Deilinyer Mendoza, 25, told reporters outside Number One Observatory Circle that he and fellow migrants did not know where they were being sent after they crossed the Rio Grande and were processed by immigration authorities in Texas. He described the harrowing trek from Venezuela to the border: hiking 12 hours a day, through the jungle of the Darién Gap between Colombia and Panama, and later traveling across Mexico, where he said migrants were faced with “a lot of corruption.”

“But thank God, we were able to make it,” said Mr. Mendoza, who arrived with his wife and was hoping to find a way to make it to family in New York. Asked about Mr. Biden, he said, “Thank God that he opened the doors to us at this moment.”

Just months after Mr. Biden took office, the administration struggled to confront a surge of unaccompanied children crossing the border, filling facilities meant to house adults beyond capacity. Months later, Border Patrol officials drew harsh criticism for their treatment of thousands of Haitians who had gathered under a bridge at the border.

Administration officials have said it will take time to unwind harsh policies aimed at keeping migrants out that were put in place by former President Donald J. Trump. It took Mr. Biden more than a year to eliminate a Trump-era program that forced asylum seekers to wait in squalid camps in Mexico, in part because of court rulings.

But disagreements among Mr. Biden’s immigration advisers also slowed progress as aides battled one another. The debates have prompted many of those key advisers to leave the White House.

Legislation to overhaul the nation’s immigration system, which Mr. Biden introduced on his first day as president, failed to gain traction in Congress.

The Homeland Security Department has been pressing the White House for months to adopt a plan that would allow Immigration and Customs Enforcement to transport immigrants released at the southern border to the cities where they wish to wait for their immigration court proceedings.

On Thursday afternoon, hours after the migrants were deposited outside the vice president’s home, ICE sent another proposal to the White House: to transport migrants apprehended and screened by the Border Patrol to cities where they have family or other sponsors, such as Miami or Los Angeles, according to a document obtained by The New York Times. There, immigration officials would finish the paperwork and processing necessary to release them to await their day in court.

The plan would also involve coordinating with nonprofit organizations ahead of time so that they are aware of pending arrivals.

The Department of Homeland Security declined to comment.

The number of daily crossings along the southern border has increased over recent weeks, a typical pattern as the weather begins to cool. Over the past few days, there have been about 8,700 crossings a day, which is historically high. The El Paso and Del Rio regions in Texas have been seeing a bulk of the crossings.

The government cannot send Venezuelans back to their country because of its lack of relations with Caracas. Instead, they are being released to one day face proceedings in immigration court. Many are given surveillance devices so officials can track their whereabouts while they wait their turn in the clogged immigration system.

The United Nations said recently that the number of Venezuelans who had fled their country had hit approximately 6.8 million, tying with Ukraine’s exodus as the largest migration in the world. Venezuela’s economic, social and democratic crisis began in 2013; economists have called the resulting migration the worst outside of war in decades.

The two private charter flights landed in Massachusetts between 3 p.m. and 5 p.m. on Wednesday. They flew from San Antonio to Crestview, Fla., in the state’s Panhandle, then to Spartanburg, S.C., and finally to Martha’s Vineyard, Mr. Freeman said. He could not explain the Florida and South Carolina stops; Mr. DeSantis’s office did not respond to an inquiry about them.

“No one on Martha’s Vineyard knew this plane was coming,” Mr. Freeman said.

In a statement on Thursday afternoon, Gov. Charlie Baker of Massachusetts, a Republican, said his state had “many resources for assisting individuals that arrive in Massachusetts with varying immigration statuses and needs.”

His administration, he added, was exploring setting up “temporary shelter and humanitarian services” at a Cape Cod military base.

As far back as December, Mr. DeSantis had mentioned the prospect of migrants arriving in Martha’s Vineyard to force a crackdown on illegal immigration. On Thursday, he said he was happy to facilitate their transport to “greener pastures.”

“Every community in America should be sharing in the burdens,” he said. “It shouldn’t all fall on a handful of red states.”

The people gathered in Niceville, Fla., cheered.

Will Sennott reported from Edgartown, Zolan Kanno-Youngs and Eileen Sullivan from Washington, and Patricia Mazzei from Miami. Reporting was contributed by Peter Baker, Michael D. Shear and Zach Montague from Washington, J. David Goodman from Houston, Jesus Jiménez and Julie Turkewitz from New York, Miriam Jordan from Los Angeles, and Remy Tumin from Edgartown.