AF-ON on D600 ... hit and miss. Me or the camera?

And I suspect you're right. I shoot sports on a D700 and use release + focus. Also, the OP should check the "Focus tracking with lock-on" setting. That's a4 on the 700. It basically sets how long the camera waits before adjusting focus in response to sudden changes in the distance to the subject.

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'You could look it up.' -Casey Stengel after James Thurber