do vampire clans in each province rival each other?
Yes. The Nine clans in Illiac Bay fought each other, but had a few allegiances. Morrowind's three clans, the Quarra, the Berne and the Aundae all fought each other, they were rivals. And Cyrodiil has one Clan, their names were lost in history but they call themselves the Order. Based by the books and the fact that they are the only Clan in Cyrodiil means they are very competitive and do not like other clans.
Read Manifesto Cyrodiil Vampyrum: http://www.uesp.net/wiki/Oblivion:Manifesto_Cyrodiil_Vampyrum
Shows that no matter how elightened and civilized another clan may be, the Order still considers them enemies to be wiped out.
EDIT: I think you meant if other clans go into other provinces. Well, I don't know about that. But it's likely since the Whet Feng can pose a threat to the Order. Also, they said they made Cyrodiil their home "As of the Third Era", meaning they probably invaded and removed any other vampire clan in Cyrodiil.