Tens of thousands of people dressed in black have marched through Mexico City in the largest demonstration yet against the government’s response to the disappearance and probable massacre of 43 student teachers seven weeks ago.

The march was marked by an outbreak of violence as a small group of protesters clashed with riot police in the city’s central Zócalo plaza. Earlier ion Thursday there had been battles between police and protesters who were trying to blockade the airport.

The disturbances threatened to overshadow the overwhelmingly peaceful march, in which bubbling anger was channelled through noisy chants that drove home the depth of the growing political crisis facing President Enrique Peña Nieto.

Demonstrators march on Reforma Avenue during a protest in support of 43 missing Ayotzinapa students in Mexico City. Photograph: Tomas Bravo/Reuters

These included counting in unison from one to 43, followed by a single cry of “Justicia.” Protesters also intoned the historical chant dating back to the dirty war against leftist dissidents of 40 years ago – “They took them alive. We want them back alive” – as well as the more combative “It was the state.” There were also constant calls of “Get out Peña.”

The students disappeared on 26 September after they were attacked and arrested by municipal police in the southern city of Iguala and then, according to the subsequent federal investigation, handed over to a local drug trafficking gang and then probably massacred. They had gone to Iguala, from their radical teacher training college about two hours drive away, in order to commandeer buses to use in a later protest. The attacks initiated by the municipal police also left six people dead.

People shout demanding the return of the 43 missing students during a massive march in Mexico City. Photograph: Dario Lopez-Mills/AP

The missing students immediately became a symbol of the country’s security crisis, which has already produced countless massacres and over 20,000 disappearances, and which protesters blame on a combination of warring criminal groups and widespread corruption and negligence that gives them free rein.

Carrying a sign saying “Mexico Smells of Death,” retired accountant Alicia Mercado said the events in Iguala had woken the country up.

“People are realising that the government is rotten,” she said, “They are seeing that the entire political class is rotten too.”

The protests in the capital were part of a day of action held across the country. The Mexican community abroad also organised demonstrations in many cities around the world amid a growing feeling that the country is facing a watershed moment.

“This movement shows that we are fed up and that real change is necessary,” capoeira teacher Alejandro Ruíz said, as he marched in Mexico City. “What we don’t know, and what everybody wants to know, is where it will end.”

The Mexico City march was headed by parents of the disappeared students, mostly poverty-stricken small farmers, but the river of people behind them epitomised the diversity of the protest movement.

Protestors burn a police car during the global protests in Guerrero, Mexico. Photograph: Anadolu Agency/Getty Images

Students, workers, accountants, housewives, performance artists, academics, and nurses joined together in the chants. The prevalence of hand-made banners underlined the spontaneity of the anger.

Some echoed common slogans, such as “I’ve had enough,” a phrase originally used by the attorney general Jesús Murillo to cut short questions at the harrowing press conference where he revealed the government’s conviction that the students were probably killed in a rubbish tip the same night they disappeared. Others were more poetic, such as “What can a country harvest if it sows bodies?”

As well as the crisis over the students, the president is also struggling to contain the fallout from revelations that a company favoured with a number of lucrative contracts by his government owns a multimillion dollar mansion specially built for the presidential family.

Rodrigo García, who works in an art gallery, said he was attending the first protest of his life because he objected to “being treated like an idiot” by this week’s release of a video by the first lady that sought to explain away the mansion as a property she was buying with the money she earned as a telenovela star before marrying the president in 2010.

“That was my limit,” the 37-year-old said, drawing a parallel between the suggestion of corruption around the house and the backdrop to the disappearance of the students. “We need to change things at their core.”

Riot police walk through a cloud of smoke from a fire extinguisher during a protest over the 43 missing Ayotzinapa students, near the Benito Juarez International airport in Mexico City. Photograph: Carlos Jasso/Reuters

Many protesters also said they were angry about the president’s recent accusations that outbreaks of violence in some of the numerous protests held since the students disappeared was part of an orchestrated effort to destabilise the country.

One banner proclaimed: “It is the not the people who are destabilizing Mexico, it is the state.”

Thursday’s march was preceded by photographs circulating on social media of youths in plane clothes travelling in army vehicles, and claims that the government was preparing to spark violence with the help of provocateurs in an effort to discredit the march.

When the battle broke out in the Zócalo on Thursday night, shouts of “no violence” and “don’t run” surged from the mass of demonstrators still in the square who had nothing to do with the scuffles.