rarely. I only moved it in this game because:
A) it was only a regent game, and I was the three-legged man in the ass-kicking contest
B) my original capital was so close to my FB that the FB was able to take care of the corruption load in my core cities enough that I could move the palace and not lose massive income
C) the bulk of my IA-aquired empire was costing me money by being just a bit too far away, and was so substantially large that I could benifit from building it into a secondary industrial base
D) I was popping GLs left and right from my military victories, ran out of wonders to hurry, so I used one to hurry a palace in a city about five cities east of my original capital (that otherwise would have taken some 100+ turns to build a new palace from scratch)
Without at least three of those four criteria, I don't think I would've moved the capital. It worked out b/c at the point the Aztecs sneak attacked me, I only had a dozen or so modern armor to rush in, and many of those were destroyed in the first wave of attacks against vet inf fortified in metros. Having the secondary industrial core allowed me to replace all those lost and add dozens more in three or four turns, not allowing me to ever weaken to the point where a couple of more powerful neighbor civs got any ideas about hitting me while I was weakened.