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And another part of the problem is that the "returning" Ashkenazi Jews aren't actually Jews. Jewishness is determined by matrilineal descent, but Ashkenazi mitochondrial DNA derives from European women.

Here we focus on both major and minor founders, with a much larger database from potential source populations. We first analyse 956 (72 newly generated) mitogenomes from haplogroup U8 (including 909 from haplogroup K, U8’s major subclade): 477 of these are from Europe and 106 from the Near East/Caucasus. We show that European and Near Eastern lineages largely fall into discrete, ancient clusters, with minor episodes of gene flow, suggesting that haplogroup K diversified separately in Europe and the Near East during the last glacial period. Of the three Ashkenazi founders, K1a1b1a and K1a9 were most likely assimilated in west (perhaps Mediterranean) Europe and K2a2a1 in west/central Europe. Most surprisingly, by analysing two new N1b2 sequences selected from a database of 278 N1b HVS-I sequences, in the context of 44 published N1b sequences24, we show that the highly distinctive N1b2 subclade, making up another 9% of Ashkenazi lineages, was likely assimilated in Mediterranean Europe, rather than in the Near East as previously proposed2. Moreover, from a survey of another >2,500 complete mtDNA genomes and >28,000 control-region sequences from Europe, the Near East and the Caucasus, in comparison with the available database of 836 Ashkenazi control-region sequences and a handful of published mitogenomes, we also evaluate the minor founders. Overall, we estimate that most (>80%) Ashkenazi mtDNAs were assimilated within Europe. Few derive from a Near Eastern source, and despite the recent revival of the ‘Khazar hypothesis’16, virtually none are likely to have ancestry in the North Caucasus. Therefore, whereas on the male side there may have been a significant Near Eastern (and possibly east European/Caucasian) component in Ashkenazi ancestry, the maternal lineages mainly trace back to prehistoric Western Europe. These results emphasize the importance of recruitment of local women and conversion in the formation of Ashkenazi communities, and represent a significant step in the detailed reconstruction of Ashkenazi genealogical history.

https://www.nature.com/articles/ncomms3543