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.

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