From what I've gathered, clan mode is like a giant FFA on Japan and the top two ranking clans for a set time period (days) move to a higher bracket or tier and the bottom four (?) move to a lower tier. The only way to conquer provinces is by fighting other players in land, sea, or siege battles. If you win you get the province, if you lose the other players (clan) get the particular province they were gunning for.
Also, is anyone else experiencing an issue with battle replays not saving properly?