Three days after Yordalis Bermudez crossed the Rio Grande into Texas, she and her family faced the same daunting question thousands of migrants do each day: what to do next.

“I didn’t have any idea where I was going to go,” said the 22-year-old Venezuelan native, who arrived with her husband and infant son.

Ms. Bermudez hopes to receive asylum, a process that can take years. Federal immigration authorities scheduled the family’s first court hearing in New York, so when officials at a border shelter said they could put her on a free bus to the city, she joined the nearly 13,000 people who have been transported to the other side of the country by state officials in Texas and Arizona.

Florida’s recent flights ferrying 49 migrants by plane from Texas to Martha’s Vineyard have generated a lawsuit, a criminal investigation and national controversy.

The human impact of that episode is small compared with the busing programs the Republican governors of Texas and Arizona have been operating since the spring to quickly transport migrants from the border states most affected by this year’s surge in illegal immigration. They initially all went to Washington, D.C., and some from Texas now go to New York and Chicago.

Migrant advocates and Democrats have accused the governors of using the migrants as pawns in a political stunt. Texas Gov. Greg Abbott and Arizona Gov. Doug Ducey have countered that they are seeking to share the burden of a record migrant surge with states and cities led by Democrats whose policies they argue are drawing people to enter the U.S. illegally.

Among those affected, the buses have proven less controversial. Several migrants said they were happy to take advantage of the easily available free rides to quickly start establishing a new life in parts of the country far from the border.

Ms. Bermudez said the bus ride was uncomfortable, with brief bathroom breaks and not enough food, but she is grateful to be in New York. The family is living in a hotel paid for by the city. Her husband is looking for odd jobs, and her son, Jean Paul, celebrated his first birthday recently with a cake brought by a member of a local church.

“At the moment, I’m content, and all’s well,” she said. “I’m happy to have a roof.”

Most complaints have come from state officials and immigrant advocates who said Texas and Arizona officials have done little to coordinate with those who can assist migrants in their destination cities.

The Republican border-state leaders said that is no different than what they deal with every day and that those who support the Biden administration should experience the effects of federal immigration-policy failures.

Texas has spent about $13 million to bus nearly 11,500 migrants out of state, while Arizona has spent some $4 million transporting nearly 2,000 people.

“Washington is the source of the problem,” said C.J. Karamargin, Mr. Ducey’s communications director. “Our message is that with this situation, the way it has been allowed to metastasize over the years, every community is a border community.”

Though the federal government regularly moves people after they enter from Mexico to process them before releasing them, the Texas and Arizona efforts are the first time states have spearheaded such an effort. It has also expanded a yearslong trend that has seen states take a more active role in immigration policy—this time by trying to shift the burden of caring for newly arrived migrants to places far from Mexico.

“This is a totally new chapter, with states going after states,” said Muzaffar Chishti, a senior fellow at the nonpartisan Migration Policy Institute.

About half the people who entered the U.S. illegally since last October have been removed, most under a Covid-era policy known as Title 42 that allows the government to quickly turn back some migrants before they have a chance to ask for asylum or some other protection in the U.S. The other approximately one million were released into the U.S. while they wait as long as several years for asylum claims to be adjudicated. They include families, unaccompanied children and people from countries such as Cuba, Nicaragua and Venezuela, whose governments won’t allow them to return.

Asylum seekers often surrender to the Border Patrol, are usually processed within a day or two, and then are released to shelters. From there they head to destinations around the country while awaiting their first court dates.

In the past, they had to pay their own fare on buses or airplanes, as many still do. When the large number of crossings have overwhelmed authorities and shelters, people have slept in tents in border cities while awaiting processing and federal agents have dropped people off in small towns that themselves became overwhelmed.

At a shelter in the border city of Del Rio, Texas, migrants deemed eligible for state-funded rides are now sent to the other side of the building where the Texas Division of Emergency Management organizes the bus trips. Most migrants who opt to board seem grateful, said Tiffany Burrow, who runs the Val Verde Border Humanitarian Coalition respite center.

“They know they have to get from point A to point B, and they weren’t quite sure how they were going to do that,” said Ms. Burrow.

She and a team of volunteers operating out of half of an old city-services building have seen about 32,000 people pass through this year, she said. Last year, about 23,000 migrants were helped at the center.

Texas began its effort in April with a pledge to bus migrants to the steps of Capitol Hill, which Mr. Abbott said was intended to send a signal to the Biden administration. Arizona quickly followed suit.

Typically the state-funded buses arrive at designated transit hubs, including the Port Authority bus terminal in New York and Union Station in Washington, D.C., where aid workers wait to help arriving migrants.

City and state officials have complained that there is no coordination with Texas or Arizona about when a bus will arrive. Instead, nonprofits and volunteers have set up informal networks to communicate about when and where buses are headed, said Johannes Favi, director of the Chicago immigrant transit assistance program of the Interfaith Community for Detained Immigrants.

Democratic officials in Washington, D.C., and Illinois have declared states of emergency.

New York has opened 38 emergency shelters in hotels. A senior administration official said Thursday that the city will erect tent cities—dubbed relief centers—to provide food, medical care and temporary housing for newly arrived migrants. Mayor Eric Adams has said that he was considering using a cruise ship to house newcomers.

“This is a humanitarian crisis created by human hands,” Mr. Adams said Tuesday. “And it was a political stunt.”

Officials in Arizona and Texas have said their busing programs are voluntary, an assertion largely backed up by the migrants who ride on them.

Andys Guerrero, a 40-year-old construction worker, and his wife, Jasmin, 44, left Venezuela in early August, traveling for weeks mostly by foot to get to Texas. Once in the U.S., they had no money to get to California as planned, so they took the free ride they were offered to Chicago.

The bus was large, new and clean, he said, with two police officers and two drivers on board. They were given a package of food for the journey, including rice and beans and nonperishable food, and rode for two days, he said.

Mr. Guerrero and his wife have been staying for the past two weeks at a Salvation Army shelter in Chicago, where they have been given new clothes and plenty to eat, he said. He is hoping to get construction work for himself and a cleaning job for his wife so they can help their children, who stayed behind in Venezuela, join them.

“It is no secret to anyone that we have been in a dictatorship for 23 years,” he said of his home country. “So the vision of the United States is a country of opportunities.”

Luis David Mantione and Oscar Jose Da Farias, also from Venezuela, left the country because of poor economic conditions almost five years ago and had been living elsewhere in South America. Mr. Da Farias said he took one of the free buses to New York to get closer to a friend in Maryland. Mr. Mantione said he ultimately wants to go to Pennsylvania.

On Monday, they were at the Port Authority terminal to pick up matching green sneakers from aid workers, a day after they initially arrived from Texas and were directed to a shelter about 1½ miles away. They said there were no beds available when they arrived, so they were directed to sleep outside in a courtyard.

“They treated us better on the bus,” Mr. Da Farias said.