Both Unite unions, T&G and Amicus DO have the kind of cross-european deals you are talking about. Both are members of a variety of bodies, not least the European TUC, which work together to stop incidences such as his from taking place. part of the formation of Unite was explicitly on the basis of becoming a union that could go europe wide, which is why there are also talk of merger with unions in germany and the US.
Union officers are also being set up in Eastern Europe as well, and they have been for the last few years. When the Single European Market came in there was a legal right for anyone to move anywhere to work, but it wasn't that widely taken up. Most Euro countries had broadly similar wages and standards of living, so the likelihood of large groups of migrant workers moving cross-borders wasn't likely to happen, and didn't. Portuguese, Greek and Irish workers did have significantly lower incomes, and there was some thought that some workers from those countries might travel, but the first two lots were always likely to go somewhere more Mediterranean than the UK, so 'we' weren't that bothered (and it didn't happen that much anyway), and the Irish had had the right to travel for ever anyway, so that wasn't any change.
It was only when the EU expanded eastwards that it became an issue. I think the number of migrant workers from those countries took everyone by surprise, and the unions weren't ready to deal with it, and hadn't thought strategic plans through enough. That was recognised during the last expansion, so more union work as done in the 'home' countries, including setting up of officers etc.
As to the EU wide union agreement no worker shall undercut a fellow worker in another EU nation, there isn't a formal one because that is a principal of trade unionism, it doesn't require an explicit document or motion saying so. Tho I am also not sure how it is relevant here, as these italian workers haven't been brought in by, or with the support of, any italian (or other) union.
