Italy is their second most popular European destination after Britain. They go there because there is nothing for them to do at home. Also, the Italian authorities seem quietly sympathetic. And because the wanderlust that created the US, Australia and New Zealand drives them too.
It's frightening even to think of the tremendous hurdles brown Asians must overcome in an increasingly racist white society. They have no education and no money. Food, clothes, lifestyle and language are totally different. They speak only Bengali. But although they have grown up at the lowest levels of a highly stratified society, they seem free of crippling complexes.
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A young man from Kishorganj took time off from selling garments by Venice's San Lucia railway station to say he had travelled all over India. He liked Calcutta best because of its egalitarian atmosphere. He didn't find there the class differences that dominated life in Delhi and Bombay.
Another young man, selling fruit and vegetables in the Rialto Market, stopped my wife from buying some cherries. "Those aren't sweet," he said, producing another lot. He was telling us about his home in Munshiganj when an American family came up to inquire about shop opening times. What's he saying? the boy asked me. His three years in Venice hadn't taught him English but he could manage some Italian. More important, he had earned enough during that time not only to repay the 12 lakh takas his family raised to send him, but also to enable his father to build a pukka house.
It's humbling to think of the courage and fortitude that underlies such achievement. There is also, of course, an element of luck. An older Bangladeshi, somewhat ragged in appearance, approached us late one evening in the Piazza San Marco, which Napoleon called the "finest drawing room in Europe", with a bunch of tired red roses. They were a euro each, he pleaded. He hadn't sold a single stalk all day. But whiners are rare. On the contrary, many young men selling tacky Chinese-made toys are defiant to the point of aggressiveness. They thaw only when assured the visitor isn't patronising.
It must be something in the East Bengal Muslim genes that makes them independent and adventurous. If one excludes the 19th century Sake Dean Mahomed who is buried in Brighton, the earliest Indians in Britain were Sylheti lascars who jumped ship to open restaurants. Today, London's third-generation Sylhetis are confident executives in banks, businesses and government offices.
The Bangladeshis I met in Venice - most were from the old Mymensingh district - may also one day become socially acceptable. But with ultra-right parties on the ascendant (the anti-immigration UK Independence Party scored notable municipal wins last week), Europe is in a frenzy about migration. Bangladeshis will inevitably be seen as a detail of the bigger challenge of refugees from sub-Saharan Africa and Syria.
It was calculated in 2010 that there were over 135,000 legal and illegal Bangladeshis in Italy. Today's figure may be over 200,000. They find Italians sympathetic. Italy faces a depression and jobs are scarce even for natives. But the police's studied indifference is interpreted as friendliness. The boys melted away as a man and a woman in blue uniform walked slowly towards us in the Piazza San Marco. The blue uniformed duo pretended not to notice.
Afterwards, one of the Bangladeshis told me that the worst to happen was confiscation of the goods they sell. Nobody is ever jailed or deported. "But a fine would mean more than five lakh takas!" he exclaimed, converting euros into his native currency.
Against that, those who have been here long enough to acquire citizenship are not just Italian citizens but enjoy full rights in 28 European Union (EU) countries. "We have a life here" they say. "A man is killed in Bangladesh for his money!" An older shopkeeper quotes Sheikh Mujib as saying that mining in other countries produced gold or oil; mining in Bangladesh yielded only corruption.
To my surprise, many of these boys say they flew in from Dhaka with visas and work permits. Surprisingly, too, they seemed to think Bangladeshis couldn't get tourist visas because the Schengen authorities demand bank certification. The rich of Dhaka's Gulshan and Dhanmondi obviously don't impinge on their thinking.
Despite the cheerfulness I encountered, an EU-funded study claimed Bangladeshis "have not been able, so far, to establish themselves in Italy in terms of finding a viable living in either business or employment… let alone remit money home." It reported that families with inadequate savings risked losing their land, pressuring the women and children to provide economic support. Sometimes, these families even have to rescue migrants from threatened destitution. Nevertheless, Bangladeshis working overseas sent back more than $10.7 billion in 2009.
As the rose seller in the Piazza San Marco demonstrated, not every seeker finds the crock of gold at the end of the rainbow. But only a braveheart and questing spirit can undertake and persevere in the search. Those who do so deserve our salute - and Dhaka's gratitude - however inconvenient some host countries might find their presence. Host countries cannot afford to forget either that free movement in a globalised world cannot be confined only to capital, goods and services.