I have a recollection that on one of the missions the camera followed the LM for quite some time

Apollo 17. Your memory is excellent. The pan and tilt rates were fixed at three degrees per second, which makes it hard to track a departing LM ascent stage. And if you zoom in too far, you run the risk of a pan or tilt shifting the object out of the frame entirely and losing it forever.

The LRV camera was online for quite some time after the astronauts' departure, and we have video of the various pans that were done on the once-again lifeless moon.