It seems that the wide availability of the Nikon D610 (and its improved shutter mechanism able to reduce dust/pollution levels to normal) makes Nikon able to extend the replacement of faulty shutter mechanisms in all D600.

Customers just have to ask. This is right in the US, but it is not clear if this is applicable in other regions.

Update: This service is now offered by Nikon France (and all other Nikon subsidiaries too).
Details on Nikon Europe.