Germany's cost-of-living protests took a sharp turn for the chaotic on Saturday night, as demonstrations in Frankfurt, Berlin, Cologne and Hamburg escalated into violence, looting and the kind of scenes that usually prompt politicians to use the word "shameful" while looking deeply uncomfortable.

Earlier in the day, thousands had gathered for largely peaceful demonstrations organised by campaign group Gen Z regelt - which translates to "Gen Z takes charge," a name that presumably sounded better before someone threw a molotov cocktail at an emergency worker. The protests targeted underfunding in schools, the high cost of living, public spending cuts and military service.

By nightfall, things had deteriorated. In Frankfurt - Germany's financial hub, because of course the bankers couldn't catch a break - police closed off a large section of the main shopping street and reported "violent riots" in the city centre, deploying special units. Shop windows were smashed, molotov cocktails were lobbed at emergency services, and a gun shop had its windows broken, with police seizing two weapons after arresting a suspect. Local media reported people breaking into other shops and stealing trainers and other items, because nothing says "fight for a fairer society" like a pair of liberated sneakers.

A police spokesperson said many arrests were made in Frankfurt. Authorities in Frankfurt, Munich and Berlin reported as many as 18,000 protesters - roughly half of what organisers had estimated on their website, a discrepancy that suggests either optimistic math or a very enthusiastic PR team. German media, meanwhile, reported tens of thousands protesting across the country.

Gen Z regelt spokesperson Lina Friedrich had framed the demonstrations as a fight "for peace and freedom, rather than for war and coercion," adding: "What matters most to us is better support for the health and education systems. We are against compulsory military service and we are in favour of a fairer distribution of the burden." Noble goals, undermined somewhat by the fireworks, stones and bottles that police said were aimed at emergency workers.

Roman Poseck, interior minister of Hesse state - home to Frankfurt - said "around 100 clearly identified left-wing extremists" had infiltrated the protest, calling the way it ended "shameful." He also invoked the French playbook: "We all have those images from France in our minds - looting, burning houses, burning police vehicles, and attacks on police officers... France must not be held up as a model in this regard." France, for its part, was busy demonstrating on the same day, with more than 320,000 people taking part in a third consecutive weekend of largely peaceful protests.

Elsewhere in Germany, Cologne police said officers near the main train station were attacked with bottles and a bicycle - yes, a bicycle - while police chief Johannes Hermanns lamented it was "deeply saddening that young people who wanted to peacefully demonstrate for their future were infiltrated today by extremists who had nothing but violence in mind." In Berlin, a smaller evening demonstration left 27 police officers injured and up to 56 people temporarily detained. In Hamburg, people dressed in black set off fireworks and torched barricades, with public broadcaster NDR reporting 120 arrests. Police said the situation in Frankfurt calmed overnight into Sunday, presumably once everyone ran out of things to throw.

The German protests coincided with larger demonstrations across Europe. In France, more than 320,000 people demanded action on the high cost of living and poor public services, especially in education, with thousands of police deployed, tear gas fired in some cities, and more than 500 arrests after clashes in Toulouse, Lyon and Marseille. French Interior Minister Laurent Nuñez said a police officer had been injured by a molotov cocktail, and Paris police said several hundred rioters attacked security forces with projectiles.

Italy saw clashes between police and demonstrators in Rome and Turin. In Portugal, police cleared a new tent camp after thousands marched against a rental bill they argue could worsen a housing crisis in a country where house prices have more than doubled over the past decade.

Mostly young protesters have taken to the streets across France, Belgium, Ireland, Spain, Portugal, Italy, Greece, Serbia and Albania. The specific demands vary by country, but the focus is largely the same: cost-of-living issues, access to housing, the quality of education and cuts to public services. Europe's youth, it seems, has found unity - just not, unfortunately, a shared commitment to leaving the trainers on the shelves.