Early EP turbos (2010) had a few serious teething troubles, particularly with timing chains and water pumps, sometimes with coolant outlets. By now all should have been replaced with the new versions of these components. It was done under warranty, and continued to be done at the manufacturer's expense afterwards. There can't be many cars not done now.
A VIN check at a dealer will confirm this work was done.
I haven't heard of a 508 tendency with this. The engine is in many models of Peugeot and Citroen and all were affected. In the small cars it's the "hot" sport option while the more modest performers can be fitted with a naturally aspirated version of the EP engine.. The large heavier cars could be got with the turbo as a petrol option. Currently the turbo is all you are offered. The design passes current pollution standards.