The European Commission has called on EU countries to take in 160,000 refugees as thousands continue to stream in.
Rallies have been held in several European and Australian cities to mark International Day of Action, demanding their governments do more to help refugees.
Saturday’s demonstrations – from Stockholm and Budapest to Athens and Madrid – came as a top UN official estimated that one million more Syrians could be displaced by the end of this year.
Yacoub al-Hillo, the UN humanitarian coordinator for Syria, said something must be done to resolve the war or the exodus would continue as tens of thousands of Europeans held rallies in both support of and opposition to refugees.
Thousands marched through London waving placards saying “Refugee lives matter” and “No human being is illegal”.
Jeremy Corbyn, Britain’s newly elected Labour Party leader and veteran socialist, drew huge cheers when he addressed the crowd from the back of a truck.
“Open your hearts and open your minds towards supporting people who are desperate, who need somewhere safe to live, want to contribute to our society, and are human beings just like all of us,” he said.
Al Jazeera’s Barnaby Phillips, reporting from the protest, said the “thousands upon thousands” of protesters were saying that Britain should do much more.
“It’s a solidarity message but the question is, to what extent does this crowd stand for Britain as a whole?” he said.
In Copenhagen, 30,000 people turned out to express solidarity with asylum seekers, while similar rallies drew thousands in Madrid and Hamburg.
A boy dressed as Paddington Bear – the marmalade-loving migrant who arrived at London’s Paddington Station from “deepest, darkest Peru” in Michael Bond’s famous books – clutched a sign saying: “Paddington Bear Was A Refugee.”
In Berlin, demonstrators waved a Syrian flag with “Refugees Welcome” written on it, while rallies in Stockholm, Helsinki and Lisbon each attracted around 1,000 people.
But at the same time, thousands took to the streets in Eastern Europe to voice their opposition to the influx, their numbers dwarfing those attending a handful of pro-migrant rallies.
“Islam will be the death of Europe,” chanted protesters at a rally in Warsaw which was attended by nearly 5,000 people and began with prayers identifying many marchers as Roman Catholics.
Hundreds also demonstrated in Prague and in the Slovak capital Bratislava, some holding banners reading: “You’re not welcome here so go home.”
The International Organization for Migration said on Friday that more than 430,000 people have crossed the Mediterranean to Europe this year, with 2,748 dying en route or going missing.
As the continent struggles to respond to the biggest movement of people since World War II, sharp divisions have emerged between the EU’s 28 member states, at both a government level and on the streets.
The “front-line” states – Italy, Greece and Hungary – are buckling under the strain.
European Commission proposals for sharing 160,000 of the new arrivals in a quota scheme are facing resistance from eastern members.
Hungary, meanwhile, is working around the clock to finish a controversial anti-refugee fence along its southern border with Serbia.
Hungary has seen about 180,000 people entering illegally this year and has passed several tough new laws that will take effect on Tuesday, meaning anyone crossing the border illegally can be deported or even jailed.
“These migrants are not coming our way from war zones but from camps in Syria’s neighbours… So these people are not fleeing danger and don’t need to be scared for their lives,” Viktor Orban, Hungary’s prime minister, told Germany’s Bild daily.
He said German Chancellor Angela Merkel’s decision to relax asylum laws had caused “chaos” and accused European leaders of “living in a dream world”.
The idea that quotas would work is an “illusion”, he said.
“The influx is endless: from Pakistan, Bangladesh, Mali, Ethiopia, Nigeria. If they are all going to come here, then Europe is going to go under.”
For his part, Austria’s chancellor has criticised its neighbour, comparing Orban’s treatment of refugees to Nazi-era cruelty.
“Piling refugees on trains in the hopes that they go far far away brings back memories of the darkest period of our continent,” Werner Faymann told German weekly Der Spiegel.
In Germany, authorities in Munich said they were overwhelmed by the flow of refugees streaming into the Bavarian capital, with more than 10,000 arriving on Saturday, and urged other German cities to pull their weight.
Germany has so far taken the lion’s share of refugees, admitting 450,000 people this year.
Merkel’s decision to relax asylum rules for Syrians has drawn praise from the refugees, but also sharp criticism from domestic allies and counterparts abroad.
In Munich, where more people were expected to arrive overnight, regional officials have sounded the alarm and urged other states in Germany – seen as the promised land by many of those seeking a safe haven in Europe – to do their bit.
“We no longer know what to do with refugees,” Mayor Dieter Reiter said, amid fears many of the new arrivals would have to spend the night outdoors.
Reiter said he was “very concerned with the developments”, noting that if other areas took in several hundred refugees it would “help to avoid chaos”.
As the newcomers arrived, some onlookers at Munich station held welcome signs to greet them.
But there were far fewer than several days ago when cheering volunteers handed out groceries and children’s toys.