I was just recently considering sending in my D600 under warranty for service. I've seen the issue, but since buying in the Spring of 2013, I've only put on about 3500 shots.
I was really, really tempted to send it in recently after hearing that some users were getting the D610 instead of having any work done on their D600 body. Something tells me they actually are tearing open these ones and replacing the entire shutter mechanism instead of sending out new D610s to these users. I probably missed the only shot at a nice little upgrade by delaying it.
Either way, I just filed for service under the service advisory.
I have heard of some users, previously sending in their D600 and actually getting a brand new shutter mechanism (last year), but that shutter was still the same exact shutter that caused the problems in the first place (it's presumed the issue is due to oil getting splattered out of the shutter mechanism itself onto the sensor).
If they are doing the same thing, and using the same class of shutter as originally found in the D600, hopefully they found a way to fix it. Otherwise, I'm not sure if it's compatible, but hopefully they are using the shutter mechanism that is in the D610. That shutter is capable of other features, ones that the D600 won't be able to support due to the rest of the body not being built around them (slightly faster burst capture framerate, and a new quiet shutter mode)... but either way, as long as it can fit and doesn't spray oil, I think that solves the real concern on the old bodies.