I will now give my analysis!
Gandalf the Grey:
He's a maiar, of the Istari, under the goddess Nienna (who, for no reason whatsoever, simply cries for a living). He happens to possess one of the (elven) Rings of Power: Nenya, the Ring of Fire, which he generally uses to "kindle the hearts of the people of Middle-Earth." It does have more capabilities beyond that, but because he's a nice person that's generally all he uses it for. He also happens to be a wielder of the Flame of Anur, which happens to be the honest-to-goodness sun (hence why he's so good at summoning light all the time). In other news, he has his magic wizard staff, which probably (and this is largely speculation here) why he can be so direct with his magical abilities. He can take down a balrog, and Tolkien even said that he could probably take Sauron in a one-on-one. In short, he's incredibly overpowered, capable of using his sheer willpower to fuel his spells.
Gandalf the White:
Alright, imagine all that, but now stack Saruman's abilities on top. Gandalf was never combative to begin with, but even then he could defeat other quasi-gods and stuff like that. He came to Middle-Earth to deal with the its people, like how Radagast was sent to deal with its wildlife. Saruman, on the other hand, was sent for the express purpose of being a military leader and combatant. He helped the blue wizards search for Sauron's spirit after Isildur made the remarkable decision to take the One Ring for himself, and then aided Rohan in destroying the kingdom of Angmar (which is why they gave him Isengard and the tower Orthanc). He is, at his core, a fighter; and now his spirit has been combined with Gandalf's. So yeah; there is absolutely no way any sucker could take him down in this form.
Albus Perceival Wulfric Brian Dumbledore:
I'm going to give this lad every allowance here because he's gonna need them if he wants to take down Gandalf. I'll make him Master of Death: with all three deathly hallows. He probably wouldn't use any of them other than the wand, though, as the Resurrection Stone isn't that useful for combat purposes and Gandalf could easily mitigate the effects of the Cloak (if Mad-Eye Moody can do it, then so can he). Dumbledore is--and pretty much always has been--the most powerful wizard in the world. There are questions of whether he'd be more powerful than any of the Hogwarts founders or Merlin, but honestly I couldn't care less because that's not the question here. He probably knows about every spell to ever exist, and can organize them via *sparkle* memory magic *sparkle* to peak efficiency. It isn't in his nature to rush through his preloaded "combos," I don't think, but he certainly can for the sake of a fight.
The Battle:
I'm going to decide that "defeat" is a broad-ish term that doesn't require any specific conditions so as not to skew this at all. My gut reaction is that Gandalf would win in either form, but I definitely think that Dumbledore could hold Mithrandir for an extended period of time. Gandalf the Grey traditionally goes on the defensive, using his powers to evade and escape (such as with the Dwarves in the goblin caves), but in times of need will definitely pull an offensive on you. His signature offensive ability is to summon the light of the sun, which he used to defeat the trolls and against the Nazgul (occasionally). Meanwhile, Dumbledore--while also traditionally staying away from battle where possible--would probably go all-out from the beginning, going from one spell to the next with ease.
If Gandalf slipped up near the start, I'd actually bet that Dumbledore could win; but if he doesn't, then Dumbledore is screwed. As soon as Gandalf realizes that this sucker's the real deal, then he'd start using his other superpowers: such as his freaking fire ring. The Elven Rings are extremely powerful; probably more so than the Elder Wand. So even if Dumbledore fights with more finesse the entire time, Gandalf can definitely win by sheer power.
In the even that Gandalf happens to be the White now, though, then Dumbledore stands no chance whatsoever. If Gandalf the White is challenged to a battle, he'll just flat-out murder you in the first fifteen seconds.
The Conclusion:
Gandalf for sure! He's a god, dangit. And he has an Elven Ring. And the power of the sun.
Also there's the issue of either of them potentially sending out a small person to do the fight for them, so... (yeah, Harry Potter would probably win in a fight against Frodo. I doubt Mithril would work against EXPELLIARMUS).