Boston’s short-term overflow migrant shelters are putting the definition of “short term” to the test.

Gov. Maura Healey implemented a cap on the state’s shelter system late last year in response to the many thousands of migrant and homeless families seeking support from state programs. These overflow sites set up across the state, including in Roxbury and Quincy, are meant to hold families for three to five days.

But those three to five days have stretched into many weeks, Mayor Michelle Wu said, as the waitlist to enter the state shelter program grows longer. That program provides private rooms and more suitable living accommodations than the temporary shelters.

“We are talking about a … rapidly changing system in terms of the number of families who are arriving,” Wu told Boston Public Radio on Tuesday.

Until more families are placed, the city is doing what it can. Knowing that the expectation of three to five days isn’t what’s happening on the ground, Wu said that Boston is immediately registering children in the shelters for school, which she said gives parents a chance to get affairs in order to figure out next steps.

“They are just trying to be safe and have a place to be able to put down roots and take care of their kids,” she said.

In response to criticisms that Boston’s wealthier neighborhoods are not taking on the responsibility of housing migrants and other homeless families in the same way poorer communities are, Wu said it has always been the case that inequities are concentrated in communities “who have contributed least to those challenges,” whether it’s affordable housing or environmental injustices.

“I think in this case, it's really a question of where does the responsibility lie for this national [and] international crisis?” Wu said.

Internationally, Wu cited the connection between increasing heat and drought from climate change that has driven some migrants to the United States. She argued that, within the United States, there are states with lower housing costs and more labor needs than Massachusetts that could be better places for new migrants to settle. The mayor said there should be better coordination between states to determine where migrants should go.

“Cities like Boston have already been shouldering the responsibility and burden of taking care of Mass. and Cass and the opiate crisis for the entire state of Massachusetts,” Wu said. Boston has vacant buildings on hand for these issues and the city is using these existing resources to bolster the family shelter system, she said.

Wu said she talks to Healey “nonstop” about the migrant crisis and how it affects communities across the commonwealth.

“She and her administration share a sense that this is something that we are having to deal with, even though it is part of a federal broken system,” Wu said. “I certainly am ready and eager to make sure that we can coordinate among even more communities statewide.”