there seem to be 2 schools of thought on the matter, either they are khazarians, or they are simply the same desert people. Personally i agree with David Duke, and his theory that the khazarian story is simply there to disarm gentiles, to get us to see (((them))) as weird eastern europeans with a different religion, when in reality they're the same sand demons they've always been, just some interbred with people of the regions they inhabited (eastern europe, north africa/mediterranean). That is also to say, that it's not some italians that look jewish, and it's not some polish people that look jewish, it's jews that look italian or polish for example.
There's also the fact that they have distinct yet obviously related phenotypes (The bear jew phenotype of epstein, from pre-slavic admixture, the rat ben shapiro phenotype from south-west slavic, the german influence in someone like ron pearlman. All of these people look very different yet they're all obviously kikes
One key element was a flip from the way genealogy was done in Polytheistic Judaism, 1st/2nd Temple Judaism, where lines of male descendants were used to show familial genealogy. Yet anyone familiar with todays Jew will know that the opposite is true today, and the rule is if your mother is Jewish that makes you Jewish..... this flip is what allowed the situation OP desceibed, with different local phenotypes infiltrating what is suposed to be an insular religious minority that prefered to live apart...