Thanks for your thoughts, all!
A few things I should clarify:
I am not moving any track out of synch w/audio. I’m just dragging the clip edge forward–thereby reducing the clip length from the left.
It’s important that I leave the layers where they are, there are too many to make them top dog for every cut.
Here’s what I did find–if I doubleclick to select the through edit points (and command+click to select multiples), I can then press UUU to get the left side, both sides, or the right side.
But after that I’m unable to do anything with it.
This simple drag forward is so basic, I feel like there must be a way to do this with multiple tracks.