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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...
 
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.
 
Hi, I was also interested in this. I just captured Constantinople as Hungary in hue old gods and is want to move my capital there because of its vast technical advancements over my current one. How do I do this (I can't seem to find the move capital button you described...)
 
It is just beside the name of the county, on the top left corner, looks like a small crown button.

As for changing the county capital, the owner of both the current capital and the one that wants to become so need to be the same. So if you want to own it yourself, but dont want to be, say, a city, you'll need to grab one of the castle within the county. The game will then switch it on its own based on what you are (Feudal=Barony Republic=City).

And if you dont want that county for yourself but dont want a wrong-type vassal, just give the county to whoever currently holds to proper type holding in that county. Example, if it is currently a city as capital but you are a feudal lord, give the city to the current baron of the castle in that county. His Barony will then become the capital on its own and he'll probably vassalize the city you just gave him.