"One good aspect of being in the EU, is that this can be done in a number of different countries at the same time"
Oh no it isn't, otherwise it might have happened before now. This 'Starbucks' situation happens because sovereign EU countries have always had the right to determine tax rates in their own country, and Holland and Ireland choice to charge corporation tax at lower rates than other countries.
The issue is then that multinationals who operate in most of these countries control the inter-company pricing regimes between each of their national subsidiaries, so that little profit is made in high-taxing countries (e.g. UK) and large profits are made in smaller-taxing countries such as Ireland or Holland.
To stop that requires enough EU collaboration within all the countries to demand transparency in inter-company pricing. Sounds fine in theory but devilishly difficult and time consuming for each national tax collector to prove.
In some markets, consumer pressure may be a more effective pressure system on these companies.