Add track select forward/backward command (A) in the multitrack like Premiere.
In Premiere, you can select everything on all layers to the right of the play-head when you select the A key and get the double arrows. (Or Shift+A to select to the left.) You can move the items on the timeline as a group.
For Audition, Jason Levine pointed out that you can use Alt-Cmd-T to select clips to the right of the play-head, but only on a specific track. I would love to move all tracks to the left of right.