The move is part of a billion-dollar scheme funded by the European Union to help border nations inundated by migrants and refugees from the Middle East and Africa.
The farewell was broadcast live on state television in Greece. Greek prime minister Alexis Tsipras posed for photos with six families from Syria and Iraq before they bordered a plane from Athens, bound for Luxembourg.
The next journey for those 30 refugees marks the start of a new program launched by the European Union. It aims to slowly relocate 160,000 people who have arrived in the continental border countries of Greece and Italy to other EU states. Mr Tsipras says it is a means of sharing what he calls a "common duty".
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"This is just the beginning. Given the thousands who have left their homes in Syria and Iraq because of war, this is just a drop in the ocean.* We know that, but we want this drop to turn into a stream, and then into a river of a fair share of responsibility."
More than 600,000 refugees and migrants have arrived in Greece so far this year.
The European Union's commissioner for migration says, for the six families heading to Luxembourg, the relocations have brought their "journey of death" to an end.
The president of the European Parliament, Martin Schulz, has told Mr Tsipras the relocation scheme can only succeed if every EU country takes its share of people.
"If we have -- I take as an example -- one million refugees, and we distribute one million amongst 500 million citizens in 28 countries, that should normally not be a problem. But if we reduce the number of countries participating in the relocation system, it becomes, for some of the countries, a problem, and I'm quite sure the problem is affecting your country in an exceptional, even dramatic, way."
The Greek Island of Lesbos says it is suffering the brunt of that problem. The bodies of at least 50 migrants and refugees remain in the hospital's morgue because the local cemetery has reached its capacity.
Lesbos mayor Spyros Galinos has proposed a registration and redistribution process in Turkey to cut the number of perilous journeys across the Aegean Sea.
"The situation is difficult because we have a strike at the harbour and there are very few departures for Pireaus, while, every day, there are new arrivals. So we have an increase in the number of refugees. So, generally speaking, the situation is difficult."
The United Nations says more children are joining the ranks of refugees and migrants streaming into Europe. Some families are taking a new route from Turkey to Bulgaria to avoid crossing the rough seas to Greece.
The UN Children's Fund, UNICEF, says at least 190,000 children have sought asylum in Europe this year: that's 700 a day out of the 760,000 people who have crossed the Mediterranean.
UNICEF's special coordinator for the refugee and migrant crisis in Europe, Marie-Pierre Poirier, says the organisation is bracing for more possible arrivals.
"The proportion of women and children was one-third at the beginning of the summer. It is now 44 per cent and continuing to increase. The proportion of children is also increasing. It was one in 10 in June. It is now one in three. So we don't see any decrease. We see an increase in the proportion of women and children and an increase in the number of children within this."
Luxembourg foreign minister Jean Asselborn says a united front is the only way forward to deal with the ongoing influx.
"Really, we have to change, very, very quickly. And if we are able to do so, I think the solidarity and the responsibility in the European Union will be tighter, will be better. And if no, if we fail, I think, and I say it, then the values of the European Union are destroyed in some way, and we have to pay attention not to go in this direction."of th
