More than 1,000 migrants have broken out of a detention in southern Mexico, one of the largest mass escapes in recent years.

While more than half of the 1,300 migrants have so far returned to the Siglo XXI facility in the border city of Tapachula, about 600 are still unaccounted for, the National Migration Institute said in a statement.

Migrants from Cuba make up the majority of those held at the centre in Chiapas state but the escapees – who broke out of the centre with no major confrontation with facility officials – also included Haitians and Central Americans, according to Mexican newspaper Reforma.

Federal police with riot shields streamed into the compound, as a crowd of Cubans whose relatives were being held at the facility gathered outside. Some of those gathered said their family members claimed poor and unsanitary conditions in the facility.

“My wife and child have been in there for 27 days in bad conditions,” Usmoni Velazquez Vallejo told the Associated Press. “There is overcrowding, insufficient food and there isn’t even medicine for them.”

The Independent has contacted state officials for a response to the claims over the conditions inside the centre.

The facility is officially listed as having a capacity of around 1,000 – which means the centre was likely filled far beyond that given that not all of the people held there escaped.

The break-out comes as Mexican officials face press from Washington to stem the movement of people towards the US border. Mexico has returned 15,000 migrants in the past 30 days, with President Donald Trump threatening once again this week to close the border with Mexico if numbers of those crossing do not fall.

Can anyone comprehend what a GREAT job Border Patrol and Law Enforcement is doing on our Southern Border. So far this year they have APPREHENDED 418,000 plus illegal immigrants, way up from last year. Mexico is doing very little for us. DEMS IN CONGRESS MUST ACT NOW! — Donald J. Trump (@realDonaldTrump) April 24, 2019

While some, including Mr Trump, have spoken about a crisis on the border, the majority of those seeking asylum in the US do so by other means. Critics point to hardline Trump administration polices, such as a tightening of asylum rules and the fact that those seeking shelter in the US must “remain in Mexico” while their cases are processed, for making the situation worse.

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In California, one of San Diego’s long-term shelters for asylum-seekers will be shutting down, saying they can no longer cope with demand in a building that is already over capacity.

“We knew that we were violating fire code and other codes by having as many people in our building,” Pastor Bill Jenkins, the director of the Christ Ministry Center, told NBC News. “But no one would take them ... We were the only place where they could have a roof over their heads and not be on the streets.”

“It seems as though [the Trump] administration is trying to make life as miserable as possible for the people who are seeking asylum,” Mr Jenkins added. “And they’re doing a pretty good job of it. I think our administration wants to prove that there’s a crisis.”

Associated Press and Reuters contributed to this report