OK so now I suspect that I have some kind of corruption going on in the project I'm trying to do this in.
I've been trying "delete hole" over and over again and finally it moved the automation. But there's a new problem. It only moved the automation if I had "move by whole measures" selected. However, with this selected, it moved everything 11 bars to the left, not 12. Without it selected, it moved all the clips 12 bars to the left but not the automation.
So to be clear, here is what's going on now
1) I have 12 empty bars at the start of the project
2) I have synth audio tracks that have automation in them, but obviously no MIDI or audio data since it's just a synth's audio output track
3) If I lasso all of the clips past bar 12 and drag them over to bar 1, everything moves except the synth automation
4) If I insert a dummy clip 12 bars long into the start of the project on one of the tracks, select it, then select all tracks, then "delete special" with every option selected and "delete hole" with "move by whole measures" selected, everything including the synth automation is moved 11 bars to the left, so I still have one empty bar at the start of the track
5) If I do the above step but without "move by whole measures" selected, everything is moved 12 bars to the left but the synth automation stays where it is.
This is all shades of messed up.