By Steve Scherer

BOLZANO, Italy May 29 (Reuters) - Some 200 mostly Eritrean migrants heading northwards from Italy were turned back by police in the Italian Alps on Friday and left sleeping in train stations as European countries tightened frontier checks before a global summit in Germany.

EU countries are struggling to stem the flow of migrants from North Africa and the Middle East and to share the burden more evenly across the continent, after more than 35,000 people reached Italy so far this year and hundreds more died at sea.

Migrants often head north after landing in boats on Italy's southern shores, hoping to make it to Germany or Scandinavian countries with stronger economies and generous welfare benefits. Some have relatives already there.

But the latest group were blocked by Italian and Austrian police as they got on trains that cross the Brennero pass to Austria, according to the migrants, Italian police and Reuters reporters who witnessed the incident.

Travellers can normally cross borders without checks in most of the European Union, but Germany has imposed controls to prevent any violent protests at a Group of Seven leaders' summit south of Munich on June 7 and 8.

That has prompted Austria to tighten controls, especially on migrants who, if turned back by Germany, might lodge asylum claims there instead of in Italy.

"We tried to get on the trains for Germany, but the police stopped us," said Josef Michael Berhan, a 40-year-old Eritrean, who slept in the Bolzano station. Berhan, who is seeking asylum in Germany or Sweden, said he is running out of money after having paid about $5,000 to smugglers to reach Italy.

EU rules state asylum-seekers must seek refuge in the nation where they enter the bloc, a policy aimed at preventing abuses or multi-country applications. But Italy and other border states increasingly say it is not fair or practical.

In a bid to help Italy and Greece, which have borne the brunt of migration from Africa, the European Commission has drafted a plan to disperse 40,000 Eritrean and Syrians to other member countries.

That has met resistance from some EU governments. Britain has said it will not participate and some eastern states are calling for a voluntary scheme.

The German border restrictions were imposed this week and are expected to last until June 16. Austrian and Italian police also called in dozens of reinforcements.

On Thursday night, 100 of the migrants slept at the Brennero station, where temperatures dropped to about 4 Celsius (39 Fahrenheit). Dozens of others curled up on the marble floors of Bolzano station, further down the valley.

"Before the migrants got through, now they don't," Fulvio Coslovi, the local secretary for the Coisp police union, told Reuters. "If they can't find a way to continue their journey, then we are going to see the numbers of stranded migrants climb quickly this weekend." (Reporting by Steve Scherer, editing by Alessandra Galloni and Mark Trevelyan)