I believe this is a very common problem with Yamahas. I had to rebuild the tranny on an 05 YZF600R that had the Dogs on the gear worn prematurely. When it pops out of gear that is what damages the Forks as well. Was a pretty easy job, but a bit time consuming.
Let me see if I can find my pics of the damaged parts and it will be self explanatory as to why it pops out of gear.
One of the solutions to this problem was to add a shift kit, or to make sure to Firmly shift all the way into gear every time. Some people are lighter shifters than others.
UPDATE:
I could not find my pictures of the damage, but what I documented was replacing a shift fork, the shift drum and 2nd and 6th gears. the Dogs on Second round off, and also where it meshes to 6th. When it pops out of gear that is when it damages the shift fork. I also bought a shift kit that had a stronger spring to make sure it goes all the way into gear. Here is a pic of what the shift kit consist of
Kit contents for YZF600r, would be about the same for FZ6
here is a pic of the damaged shift fork.
You can see here the Dogs are worn on the edges. This gear looks just like the ones in our FZ6's (3 dogs)
Some manufactures use like 5 or 6 to avoid this from happening so easily.
Please note these pics are from a YZF600R rebuild, not he FZ6!!!
But should be about 98% the exact same.
Todd