I confess to having read the 13th Tribe many years ago. There are Khazarian descended Jews who are quite proud of it (they have web sites) and Israel has spent a good deal of public money on the research. It is a given that the mass nobility and what would have passed for bourgeoisie converted to Judaism but the debate is over how many of the lower sectors of the vast Khazarian empire did.
A fascinating study for me.
And yes....I know Koestler was a bit odd. My hunch is that the Khazars bumped the Ashkhenazim populace markedly...but that is only a hunch.
Actually, from what I've read, the Pushtins of Afghanistan/Pakistan fit closest to the lost tribes. They do so both in naming conventions, in culture (except that they are islamics now) and in folklore. They say they were placed there by the Assyrians as a punishment. Also, there is a jewish black tribe (proven genetically by the priest gene) stretching from eastern S.Africa through Mozambique.
The Khazars converted and are an asiatic people. I've seen some before and they look nothing like semites but a lot like eastern asiatics.
Before the Khazar Khaganite:
Rise of the Khazars. Note they conquered Kiev which from around 700s was a slavic city.
Height of the Khazar (note the location of te Bulgars!)
Decline of the Khazars (Note the Magyarian migration into modern day Magyaria/Magyarorszag/Hungary)
Unfortunetly for the Kieven Russ, the conquest of their enemy/competition, the Khazars, unleashed the Pechenegs who were in turn keeping the Mongols in check.