How to move your capital?
1. How to move your capital from one province to another?
2. How to move the capital from one barony to another within one province?
I did not use it, but I did see a small "move capital" icon in a non-capital demesne province at the top of one of the province screens (both were in Italy, not far apart).
If the current county capital is the wrong type (temple or city) and you acquire one of the good type that one will become the capital.
I do not know how to change it otherwise.
I'm also interested on this!
I started playing as king of Galicia, but for some reason my capital moved to Lisboa when I liberated the county from the muslims...
Now I actually fancy moving it further south, but can't really find a way to do it...
That guy with the thing
As Sir Garnet said, there's a "move capital" button in the province view of each of your counties that's not the capital.
It's in the same place the "this is the capital" picture is in your capital.
AFAIK, you can't purposefully change which settlement in a province is the county capital.