Does this have to be tied to Euro membership?
I don't think so. Strong fluctuations against the Euro would be bad, however. One possibility is to fix the income in Euro and pay it out in the currency the receiver likes.

Are such transfers as will occur inflationary for a country like (say) Slovenia?
Probably yes. But with free movement of goods and in not so distant future free movement of people, wouldn't the prices anyhow have a strong bias to the 'general EU price level'? Of course you have some issue with too much people unwilling to work, when you have a basic income, which is initially more than basic in the poorer parts of the EU, but I think with a moderate income taxation starting at the first cent (or whatever they have there) earned, it should have not too dramatic effects.

Whilst pushing a basic income out across the Schengen zone solves the "freedom of movement in the EU" problem, is there a danger it will greatly amplify problems at the borders of the Schengen zone?
So this would be Switzerland and Norway, right?
I think they are small enough and rich enough, to introduce only a managable immigration stream, if at all. Already now Switzerland is a tax haven, and even inside the EU there are vast differences with inheritance tax, income tax, corporate tax. The last in my view a bigger distortion than what the basic income introduces with respect to Switzerland and Norway.

Der Amerikaner ist die Orchidee unter den Menschen
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