I tried everything on the edit menu but can't seem to drag a component such that the connected buses follow "nicely". Altium essentially only rubberbands the last trace segments instead of the whole trace.
You have to set your preferences to allow dragging tracks with a component. Go to Preferences>>PCB Editor>>Other>>Comp Drag, and select "Connected Tracks". Now when you use "move component" (shortcut MC), the tracks will drag with the component as long as they aren't locked.
Note that it is only the tracks connected to the component that drag. The entire net doesn't drag - that would make a mess of the board.
I left out one pointer - the "Other" settings are on the "General" page of the PCB Editor Preferences. My post should have said "Preferences>PCB Editor>General>Other>Comp Drag". That setting has been in the preferences pages of every version of Protel/Altium as far back as I can remember.