To move all the clips with automation (so the envelope events stay at the correct spot on the clips) I think you actually have to use the Edit > Copy/Cut Special function. Otherwise the clips will move but the automation will stay put.
So select the folder (which will select everything inside the folder) > Edit > Cut Special (in the Cut Special dialog make sure automation is checked) then paste at the new time (I think you have to set the Now Time for the start of the selection to land where you want).
I've never actually done this so I may be wrong. If it was just the clips with no automation (or if the automation is on the clips/not in lanes) then just selecting and dragging would work.