Who's the "they" you're referring to? Most Israeli Jews are Mizrachim, not Ashkenazim. Either way even if Ashkenazim do 100% originate from Khazars, it doesn't matter about who is making what claims: the fact is that most Israeli Jews living there have been born there.
Shaffir to Herzog: Opposition failed to show leadership in political battles against PM (Haaretz)
from the article: The Jewish people's ultimate treasure hunt