I would probably go local. You could even have it be the same character. A elf strong in the use of magic can live 1,000 years.
One problem with that: the Champion of Cyrodiil eventually turns into a daedra, and makes an appearance in Skyrim. So huh, not really logical.
I'm also playing with the idea to replay all the games, but then as close to the main quest story and lore as possible. The Nerevarine would be a male altmer (they look most like the chimer after all, and I think it would be wrong for the Nerevarine to undergo the same curse as the Dunmer), the champ of Cyrodiil a male Breton (well, he/she can be anything really, I'm simply basing this on the daedra he'll become), and the Dovahkiin a Nord. Heck, they still could be descendants of each other, the children inherit the race of their mother in Nirn anyway.
Although, playing a super old elf (you can play a young character in both Morrowind and Oblivion, to become more aged in Skyrim) would be cool too, like that-guy-again said. You can imagine your character saying in-game "Yeah yeah, I know about that Nerevarine and Champion of Cyrodiil stuff, I'm them after all". And heh, you can also just imagine you made a friend from Cyrodiil replace you as the daedra temporarily, so he can fulfill the quest in Skyrim. ^^
Don't need no dunmer or altmer for that
Spoiler the Nerevarine becomes ageless due to corprus.
Heeehhh. O_o I must've completely forgotten that. Now this becomes quite interesting.