Germany's southern city of Munich was expecting 10,000 migrants to arrive by train in a single day, with officials saying that 40,000 in all could arrive in the next few days.
By this morning in Munich, 3,600 had already arrived.
Chancellor Angela Merkel, facing criticism at home, said taking in asylum seekers was "the right thing to do" even as her deputy warned that the country was "reaching the limits" over how many refugees it could take.
The latest wave comes two days ahead of a European Union meeting in Brussels on Monday to discuss the migrant crisis triggered by war and economic privation in places such as Syria, Iraq, North Africa and South Asia, and a reluctance by many countries to allow any more in the door.
The Washington Post reports that thousands of protesters in some parts of Europe have turned out on the streets in central London, "some with signs that said 'Reject the Politics of Fear,' according to The Guardian.
Meanwhile, in Prague, thousands gathered for anti-Islam rallies targeted at the migrants, mostly from Muslim countries.
Hundreds more reportedly came out in Budapest, where many migrants board trains for Germany, to show their support.
The Associated Press reports:
"Some 450,000 migrants have arrived in Germany this year. The country is expecting at least 800,000 in 2015 — by far the most in the 28-nation EU.
"The welcome mat for migrants in Germany contrasts sharply with the dangerous seas they faced to get to Greece, the tear gas from Macedonian police, the razor-wire fences, crude migrant camps and burly police in Hungary and, at times, train service shutdowns along the route. ...
"While Germany has largely welcomed the migrants and a poll this week showed broad support for the government's course, Merkel has faced some criticism from Bavaria's governing party — the Christian Social Union, which is the local sister to her own conservative Christian Democrats."