This has been bugging me for a while. For slavers to raid Elseweyr, they'd have to either cut across southern Cyrodiil or take a long path deep through Black Marsh.

I just don't see how this is possible in such large quantities. Most if not all raiding parties wouldn't be allowed to travel through Cyrodiil with their "merchandise", except for smuggling. And I doubt more than a few very lucky Dunmer would survive a round trip through Black Marsh, especially not slavers.

So how do the Dark Elves get Khajiiti slaves?

Maybe the khajit that went to trade in morrowind got captured?
Or they went in large numbers to get loads of them from Elsweyr

They'd be smart enough to stop coming after the first visit.
As for your later suggestion, that's more akin to an invasion, which would DEFINITELY not make it through Cyrodiil or Black Marsh. Perhaps by sea, but I can't think of anyone in that area of the Tamrielic coast that would allow slavers to dock at their ports.