Barely two weeks had passed since the migrant crisis arrived in their city of 40,000 people, 10 miles northwest of Boston, but the volunteers gathered at a church in Woburn on a recent evening sounded battle weary.

The small group of locals — including a kindergarten teacher, a Methodist pastor and a Haitian American woman who works in health care — had stepped up to help some 80 migrant families whom the state placed in Woburn hotels last month. Determined to offer a warm welcome, the volunteers had quickly discovered the daunting complexity of meeting basic needs, a reality check also underway elsewhere in the state and nation.

In Massachusetts, the only state with a right-to-shelter law that guarantees every family with children a place to stay, the crisis has been accelerating, with more than 80 cities and towns receiving migrants to date. (New York City has a similar law that has generated tumult and debate and that goes further, requiring it to provide shelter to anyone who asks.) The number of families living in emergency shelters and hotels statewide has doubled in the past year, to nearly 6,300 last week; the cost has ballooned to an estimated $45 million per month.

Officials estimate that as many as half of currently sheltered families are recently arrived migrants from other countries; most have come from Haiti, drawn by word of mouth and the pull of the state’s well-established Haitian community.

Gov. Maura Healey declared a state of emergency on Aug. 8 in a bid for federal help, joining New York, Chicago and Washington, D.C., which have taken similar steps. On Aug. 31, Ms. Healey authorized more than 200 National Guard members to assist the more than 2,500 families living in hotels, a step meant to address a shortage of social service agencies to help incoming migrants.

The state is also housing migrants on two college campuses and on a Cape Cod military base, and has opened two welcome centers to process arrivals, with many coming from the southern border, advocates said. At one of the centers, in Quincy, south of Boston, most of the nurses and caseworkers are Haitian Americans who speak Haitian Creole, allowing them to quickly identify needs and brief families on next steps.

On average, 10 new families show up at the center each day, staff members said. The goals are to assess their health, help them set short- and long-term goals, sign them up for key state services, and move them to housing elsewhere, all within five days.

In Boston and its suburbs, on Cape Cod and through much of the less populous western half of the state, the rapid influx has stressed local social service agencies and volunteer resources, and heightened political tensions, even among the state’s many liberal-leaning officials and residents. Many costs, like hotel fees and meals, are covered by the state, which also plans to reimburse local school systems at a rate of $104 per student per day.

Still, some local leaders say the placements are inequitable, skipping over towns without hotels or shelters, and calls have mounted for more planning and advance notice.

By Friday, the number of families placed in Woburn hotels had reached 150, said Mayor Scott Galvin, a seven-term Democrat seeking re-election to the nonpartisan office. He said the situation was not sustainable, and called for state legislators to consider changes to the 40-year-old right-to-shelter law, which he said was “passed at a different time, and was not meant to cover what we’re seeing now.”

“We’re going above and beyond, while some communities around us are not being impacted, and we don’t have endless capacity in our schools,” he said in an interview. “The benefits that are bestowed on migrants make the state a very attractive destination, and without some changes, this challenge is not going to abate.”

Kelley Hurley, a Woburn teacher, said she saw an opportunity in the migrants’ arrival, to help nudge her changing city toward a warmer embrace of its new diversity. She had observed the trend for years in her kindergarten classroom, where her students spoke eight languages last year. But as she revised her own curriculum to reflect the shift, she worried about stubborn pockets of resistance in a place long defined by its white Irish Catholic and Italian heritage.

Woburn’s 4,300 public school students were 65 percent white last year, compared with 76 percent in 2013 and 86 percent in 2003.

“It felt like the city hadn’t quite caught up with the changes, and I thought, ‘Here’s a way to get people involved and excited about it,’” Ms. Hurley said.

She quickly found eager partners at two local organizations, the grass-roots Woburn Welcomes and the nonprofit Social Capital Inc., and at several churches. Donations of money, clothing, diapers, toys and car seats flooded in, while two local women with Haitian roots pitched in as translators and fixers.

Strangers before mid-August, the volunteers were now close-knit allies exchanging dozens of daily text messages. They had struggled to transport dozens of newly arrived migrants to laundromats, chafed with frustration when meals for the families arrived late from a state-contracted company, and seethed when a hospital asked for a $300 deposit from a migrant woman suffering a miscarriage.

“We wanted to take care of the families, and we’re still doing that,” Ms. Hurley said. “But we’re trying to be realistic.”

At a recent meeting where volunteers brainstormed ways to transport families to a free church thrift shop, they also resolved to clarify what services the state planned to provide. The intent was to tailor a more sustainable role for themselves, one they could balance with their regular jobs.

“We don’t want to burn people out,” Ms. Hurley added.

She noted, with some pride, that the state had asked the Woburn volunteers to advise like-minded groups of citizens in other towns. But as she returned to her full-time teaching job last week, she worried, too, that her group’s exhaustive efforts might lead the state to think they could persist without more help.

While the outpouring of support has made them hopeful, the volunteers said they were avoiding Facebook, where some other Woburn residents have railed against the local migrant placements and questioned why outsiders should receive free shelter.

Late last month, about 20 people staged demonstrations outside several Woburn hotels housing migrants, with a banner that included the name of a neo-Nazi group. On social media, the hate group described the action as an “emergency mobilization” to “oppose invaders and their collaborators.” Participants came from several states, according to the police.

No one was hurt or arrested. But volunteers who translate for the migrant families said some had been afraid to leave their rooms after the demonstrations.

In general, the increasing number of migrants in Massachusetts has given rise to less resistance than in some other places, like New York, which saw hundreds of people protest an emergency shelter in a former Staten Island Catholic school last week. The city is now sheltering more than 50,000 migrants nightly, while many upstate counties continue to fight attempts to place migrants there; Gov. Kathy Hochul has resisted calls to force the issue.

With Massachusetts still seeing far smaller numbers of arriving migrants, a spokeswoman for Ms. Healey continued to call for “the partnership of communities to ensure that families have a safe place to stay.”

The state, one of the most affluent and politically progressive in the nation, was among the first to be targeted by Republican governors seeking to force Democratic leaders far from the border to confront the immigration crisis. After Gov. Ron DeSantis of Florida shipped two planes full of migrants to Martha’s Vineyard, off the coast of Cape Cod, last fall, residents of the island rallied to their aid.

Maine, too, has developed a reputation for helping migrants, fueling a stream of new arrivals, many from Africa, that has overwhelmed the state’s scarce housing stock. In Portland, Maine’s largest city, 200 asylum seekers lived in a sports arena, the Portland Expo, for months this year before they were recently moved into hotels.

For migrant families, tenuous housing situations, combined with long waits for work permits, create a mix of gratitude and anxiety. On the sidewalk outside one of the Woburn hotels last month, where Haitian migrants socialized in the sunlight, one 16-year-old, a fluent English speaker, said he was excited to start 11th grade after missing seven months of classes while in transit to the United States.

Translating for several adults, including his father, the teenager said their most pressing concern was how to swiftly become authorized to work. Current rules delay asylum seekers’ ability to work legally; Ms. Healey and elected officials in other states have increased pressure on the federal government to revise those policies.

“The problem is that they want to work, but they have no card to work, and it takes too long,” the 16-year-old said, summarizing the concerns of his elders.

Schools have scrambled to accommodate new students. After enrolling more than 50 new students from migrant families in August, Woburn school administrators asked the city’s school committee for last-minute funding late last month to hire six staff members.

Matthew Crowley, the superintendent, acknowledged that more staff might be needed once the students’ past trauma and emotional needs are assessed.

“It’s a real thing we’re going to have to unpack,” he said.

John Wells, a committee member, proposed a letter to Woburn’s state representatives “acknowledging our eagerness to help people in need, and our need for funding.”

“We’re taking up slack for other communities,” he said, “and we need the funding to do that.”