By the time of my visit, the broader question in Hungary and throughout the member nations was whether the European Union itself was falling apart. Twice in 11 months, Paris had been hit by vicious terrorist attacks, raising fears about European security and stirring anti-Muslim xenophobia. Europe’s far right was gaining strength, including the National Front in France. Europe’s undisputed leader, Chancellor Angela Merkel of Germany, was in political trouble. Greece was still a financial mess. A demographic crisis loomed, given the rapid aging of many European countries. Out on the European periphery, Russia’s president, Vladimir Putin, was making bloody mischief in Syria and Ukraine. And the United Kingdom was contemplating withdrawing from the union altogether.

The European Union was supposed to be an economic superpower, but after seven years it is still struggling to recover from the global economic crisis. Economic growth is sluggish at best (and uneven, given the divide between a more prosperous north and a debt-burdened south). Adjusting for inflation, the gross domestic product of the 19 countries now sharing Europe’s common currency, the euro, was less in 2014 than it was in 2007. Widespread joblessness and diminishing opportunities confront an entire generation of young Europeans, especially in Spain, Italy, France and Greece. The economic malaise tinges everything: Young people resist marriage for lack of economic opportunity. Poorer European countries are experiencing brain drains as many of their best young professionals and college graduates move abroad. Numerous Greek doctors, for instance, now work in more prosperous Germany while Greece’s health system is in crisis. Even as Toroczkai pushed back against migrants, he complained to me that too many young Hungarians had to leave for London or elsewhere to find work.

The migrants only accentuate the European paradox: A place of deepening pessimism for many of its own young people has become a beacon of hope and safety for migrants, many of them Syrian refugees who have been through the horrors of civil war. Many are young and educated, seemingly a timely fit for a region with an aging population. Except Muslim immigrants present a challenge to European ideals of tolerance, especially in a year of terror attacks, as far-right extremists and conservative political leaders like Orban warn that Europe’s security and ‘‘Christian values’’ are threatened — a reminder of just how fragile the European system has become.