Hi,
I've just had a go at this.
I've got an Instrument track which contains MIDI. Associated with that track are three Automation Lanes, one for Pan, one for Volume and one for a parameter withing the soft synth.
If I simply grab the clip to move it, the automation moves with it, all three take lanes.
You will notice, in track view, that when you select a track, the track number background turns blue, and so do the take lane selection indicators.
What I tried next was deselecting the take lanes by Ctrl clicking the take lane selection gizmo. I can either deselect all three tracks or just one or two. Which ever tracks are deselected, the automation stays where it is, whereas the take lane that is still selected will have its automation moved along with the clip.
I hope that helps. I will see if I can find the relevant pages in the X2 Reference Guide which confirms the above.