Looking for RP reasons a race other than Nord would be in Skyrim and be willing to purge the land of its evil denizens. It's easy enough to justify if you're RPing a treasure hunter, but what if you're not? Why would a Breton care about some Draugr-infested dungeon in Skyrim? Why would an imperial be doing anything other than hunting down Stormcloaks?
I'm coming a bit late to this thread, but I'm going to approach your question from a different angle. Remember, just because the majority of the natives are Nords, that doesn't mean that other races weren't born in Skyrim. Take a look at the Jarl of Whiterun. His top housecarl is a Dunmer. His steward is an Imperial. That Imperial's daughter? Runs the blacksmith shop near the city gates, and she married a Nord whom (I think) is the guy behind the counter in that shop. The fletcher shop across the street has a Dunmer mercenary. She seems to have been born in Morrowind, but as a merc she probably also qualifies as an adventurer of sort, so right there is a motivation. Money.
To approach it from the way you probably wish people to answer, here's some reasons:
Dunmer: Daedric worshipper, but favors Meridia. Meridia despises undead. Thus said Dunmer wants to clear any dungeon filled with undead since they are an abomination in his eyes. Necromancers might seem just as bad. Also, remember there's Dunmer in Cyrodiil whom maybe have never set foot in Morrowind, and think of themselves as Imperial citizens first, Dunmer second. That sort of Dunmer may have followed north to Skyrim to help put down the civil war, how dare that Stormcloak faction try to split from an empire that is already weakened.
Imperial: The dungeons and ruins often contain undead. If those undead got the bright idea to leave the ruins and invade the surface, it would be just as bad if not worse then the civil war. Thus the Imperial wishes to clear dungeons to prevent that. There's also bandits about, possibly due to the civil war making it harder to patrol the roads. The Imperial wishes to clear out the ruins of bandits so that the roads in an Imperial province are safe. Or maybe so that his fellow Imperial legion soldiers can travel without having to worry about ambush from said bandits, worrying about ambush from Stormcloak rebel soldiers is bad enough. Again, someone of the Imperial RACE might have been born in Skyrim though, and thus may just in his eyes be wandering his native land, even if his race is a minority in said land.
Khajit: We already see them in tents outside of many of the cities as what seems to be nomadic traders and merchants. Thus a bit of wanderlust is apparent in the race. One may have a bit more wanderlust and want to do more then just be a nomadic trader. Or maybe he's sent by his clan to gain odd artifacts from a distant land like Skyrim. Or maybe he's out to seek his fortune so he can form the nomadic tent caravan of his dreams, so he travels to Skyrim to explore it's tombs and ruins to gain his fortune, so he can come back home wealthy.
I'm pretty tired, had insomnia and was up much of the night, going to finally try to sleep now. So I'll stop with those three races. But really, just take a look at how each race seems to act, and go from there. I actually don't know a lot of Elder Scrolls lore, so I'm sure someone with much more lore knowledge can possibly come up with better ideas.
(As an aside, I do know that the idea is that the character was captured after crossing the border into Skyrim, and that if one isn't a Nord, the Imperial officer says "Oh, I'll return your remains to
". But maybe he's just assuming about the province..he might be thinking "Oh, Dunmer..Morrowind" and never bothers to ask if the guy is from a different province, or was even born in Skyrim. And maybe you were crossing the border after a long trip to get back HOME to Skyrim. Or maybe you were just caught near, and they assume you were crossing into it since you're a minority race in Skyrim)