According to NR, there are several Nikon D600 users are able to get a new Nikon D610 camera from Nikon as a replacement for their oil/dust issue D600. This is a good news for D600 users.
In some cases (I believe it was France), a small fee was charged for getting a brand new D610 camera.
In United States:
Similar report from the US – see the above screenshot. I have no idea how Nikon determines who can get their D600 replaced with new D610 camera. Maybe if you use it professionally or if you have sent it already too many times for sensor cleaning – I really have no idea. The standard procedure for the past few months has been for Nikon to replace the shutter mechanism in every D600 they receive.