My plan is to have the descendants of and new members of the Knights of the White Stallion be on an expedition into Skyrim under order of the current Count of Leyawiin. Do they still use the Count system? Do we have a name for the current Count?
The only problem I can see with that is that The Knights of the White Stallion would likely have been eradicated/ forced to disband during Thalmor occupation, assuming they even survived Leyawiin becoming a city-state then being reabsorbed. I suppose you could have a descendant discover his origins and reforge The Knights of the White Stallion ala Knights of the Nine. Whatever suits you, I don't think there's anything definitive regarding them.
If you wanted you could have them falling back to the imperial city to rally with Titus Mede II's forces when the Thalmor attack, then being part of the story from then on.
Leyawiin also likely still has a count, but I'm not sure if its ever mentioned, and if it is the count/countesses name.