Flimsy boats packed with migrants continued to land in Greece from Turkey on Sunday despite the start of a landmark deal between the European Union and Ankara to stem the massive influx.

Under the controversial deal, which came into force at midnight, all migrants landing on the Greek islands face being sent back to Turkey.

And in a grim start to an agreement designed to stop people from making a journey fraught with danger, two little girls were found drowned and two Syrian refugees died of heart attacks after the crossing.

Nine more died and hundreds were rescued off Libya, the Red Crescent said, as fears grow that the shutdown of the Greek route could encourage more people to attempt the even riskier Mediterranean crossing to Italy.

On the Greek island of Lesbos, police said some 800 migrants had arrived by midday despite the EU-Turkey deal formally coming into effect.

Officials said it would take time to start sending people back, as Greece awaits thousands of European staff needed to take on the daunting task of mass repatriation.

The SOMP agency coordinating Athens' response to the crisis insisted however that those arriving from Sunday faced certain return to Turkey.

‘They will not be able to leave the islands, and we are awaiting the arrival of international experts who will launch procedures for them to be sent back,’ the agency said.

The European Commission has said the agreement, which has faced international criticism, will require the mobilisation of some 4,000 police, security staff and other personnel.

France and Germany have offered to send up to 600 police and asylum experts, while Romania said Sunday that it would send 70.

- Thousands protest across Europe -

Under the deal, for every Syrian among those sent back from Greece to Turkey, the EU will resettle one Syrian from the Turkish refugee camps where nearly three million people are living after fleeing their country's brutal civil war.

The EU will also speed up talks on Ankara's bid to join the 28-nation bloc, double refugee aid to six billion euros ($6.8 billion), and give visa-free travel to Turks in Europe's Schengen passport-free zone by June.

The aim is to cut off a route that enabled 850,000 people to pour into Europe last year, fleeing conflict and misery in the Middle East and elsewhere.

But Amnesty International has called the deal a ‘historic blow to human rights,’ and on Saturday thousands of people marched in European cities including London, Athens, Barcelona and Amsterdam in protest.

On Lesbos, Gatan, a Syrian who had just arrived with his wife and two children, said he chose to ignore warnings about the deal.

‘In Turkey they told us not to go to Greece, that we risk arrest,’ he told AFP.

But he added: ‘We could not stay in Turkey. We want to go to Germany or France.’

- 'Treated like animals' -

Realistically, migrants will likely not start being returned to Turkey until April 4, according to German Chancellor Angela Merkel, a key backer of the scheme.

EU officials have stressed that each application for asylum will be treated individually, with full rights of appeal and proper oversight.

The deal also plans major aid for Greece, a country now struggling not only with a debt crisis but with some 47,500 migrants stranded on its territory.

On the Macedonian border alone, 12,000 people are stuck in dire conditions, after a string of border closures on the Balkan migrant route that has left Athens dealing with a huge bottleneck.

The charity Doctors Without Borders said 33 migrants had been treated Sunday for injuries inflicted by Macedonian police as they tried to slip across the shuttered border with Greece.

In Turkey, around 200 migrants were caught off the coast and turned back as they tried to reach Greece before the deadline, a Turkish coastguard official said.

They were detained on a ship for 36 hours and were being held in a pen in the port of Dikili, where they pressed against the metal gates, chanting ‘We want to go! We don't want to stay!’

‘We were treated like animals on the big ship. They gave us no food or water,’ a 19-year-old Syrian told AFP.