This issue was not born just now, the biggest jump was last year when first Italy and then European countries decided that the boats should be rescued right out of libyan territorial waters because of a few sinking, thus causing an enormous increase of illegal immigration through increasingly more dangerous boats (actually dinghies) with the final result of even more people dying in absolute numbers. Actually the real origin is the destruction of the Libyan state since the problem was of a controlled size before then.
The Syrians suddendly flooding through Kos island is only an added problem on top.
Problem
- welfare shopping: asylum seekers want to ask asylum in Germany because they have the best chances there, not because other countries in the Dublin zone are unsafe.
- acceptance rates: different countries have different acceptance rates, so asylum seekers pick the country with the highest rates for people from their country (not a problem for Syrians).
- abuse of asylum: economic illegal immigrants (kinda like your Mexicans) ask for asylum and make up stuff (e.g. claim they're syrian or that they're politically persecuted in countries not at war) just to live a few months on the dole during processing, maybe slinging drugs in the meanwhile and to get a chance to disappear and live as an illegal in the country (in case the request doesn't get accepted).
- the Dublin system: if you get registered/ask for asylum in one country in the dublin zone, if you run to another (richer) country and ask for asylum there you will be sent back to the first country. So nobody wants to be registered in Greece, or Hungary. That's why there's this mess in front of the train station.
- Germany and Sweden are getting too many asylum seekers, Eastern Europe likes the fact that nobody asks for asylum in their countries.
Hungary is respecting the EU rules, Germany is not because they decided that they're not going to send any syrians back to Greece. This caused the current mess in Hungary.
So now everybody on the Balkan route is suddenly a syrian.
An additional note: most on the balkan route are legitimate refugees. But they were in Turkey before, which is a safe country. They're asking for asylum in Europe because of the better long-term chances. There is nothing that can be done about this, we can forget about a worldwide refugee distribution system until there is a world government.
The point is, these people would have not left Syria if there wasn't a war.
On the balkan route there also are people from the indian subcontinent (mostly bangla), these are not legitimate, but the bangladeshi have a very organized network for illegal work so they don't care about getting asylum, just about getting in.
Those who arrive in Italy are in many cases not legitimate. West Africans are not legitimate asylum seekers and often commit crimes.
Eastern Africans also move for economic reasons, but Eritrea punishes emigration without permission as desertion so they cannot be sent back as that's a serious violation of freedom. Checkmate.
Solution
Dublin-wide asylum application in the first country of arrival with uniform acceptance rules, common-funded processing centres in those same countries, quickened processing so as to immediately send to expulsion centres in airports the abusers, then redistribution of the asylees and refugees to every country according to the population. Refugees should be sent back once the war has ended unless they're a net contributor (i.e. they don't need welfare).
Other kinds of asylees are probably going to stay forever.
If you are assigned to a country, you have to stay there and you will only be able to collect welfare and get housing if you stay there.
If there's too many people and the time needed for processing can't be reduced, also asylum seekers and the processing centres should be distributed equally among the countries.
Additional advantage: the chance of ending up in a poorer European country and not being able to choose it in any way by not getting identified by police will decrease the allure of illegal immigration for africans.
So doing this will help Europe long-term.
Political obstacles