I want the background to move as the character walks. My question - is there a tool in Character Animator for camera pan or trick?
You can vote for the camera-pan-as-a-feature... :-). Camera panning support – Adobe video & audio apps
Until then, the background is just a puppet. Anything you use on normal puppets you can do to the background as well. You can drag it or (what I normally do) record a new Position X value for it then use a blend to smoothly transition from the old to new X value. I also do it for Scale values. See Simulated Camera Panning in Adobe Character Animator - YouTube for a quick video.
There was a tips and tricks video a while back that had another rather cool approach. They used the Walk behavior on the character puppet with body movement speed zero (so legs would move, but puppet stayed in middle of scene), and put another walk behavior with negative body speed movement on the background. So just using the walk left/right arrows would move both the character puppet and the background at the same time - looked very cool! (They went further and put another furniture layer in front that moved faster than the background, giving a parallax feel.)
Yes! That was it! Thanks for the video link!