That's a really good question, and even just a few years ago we didn't really have anything approaching a conclusive answer.
However, an analysis of mitochondrial DNA extracted from human remains recovered from the Xiaohe cemetery was published in 2015, and we now know that the answer is complicated, but that most of the early populations probably arrived from the steppes of southern Siberia at some point in the early Bronze Age, rather than as European settlers travelling the Silk Roads to China.
As you quite rightly said, since the Tarim mummies were "re-discovered" by the Xinjiang Archaeological Institute in 2000, two main theories have developed to explain their origin.
The first theory suggested that the earliest settlers in this region were nomadic herders who had migrated from the steppes of Russia, Siberia and Kazakhstan. The other theory suggested that people first travelled to the region from Bactria (roughly, modern Uzbekistan, Afghanistan, and Turkmenistan).
Unfortunately, the archaeological evidence was inconclusive. The extremely arid conditions resulted in unusually good preservation of organic remains, so we really do have a lot of evidence. In cases like this we usually rely on similarities between the archaeological evidence from different sites to suggest links between ancient peoples. This might include burial practices, the types of animals being farmed, or even (where it survives) clothing. In this case, we have all of the above, but the evidence was inconclusive. Some evidence supported one theory, and other evidence (from the same site) supported the other.
Similarly, earlier attempts to determine the genetic origins of the population of the Tarim Basin, based on human remains recovered from the Gumugou cemetery were also inconclusive.
The investigations were the subject of a National Geographic program titled China's Mystery Mummies, filmed in 2007.
Recent DNA Analysis
In a series of papers from 2010 to 2015, Chunxiang Li of the College of Life Science, Ancient DNA Laboratory, and Research Center for Chinese Frontier Archaeology, at Jilin University, together with a number of colleagues, published a series of papers that presented the DNA evidence recovered from human remains from a large number of individuals from the Xiaohe cemetery complex. That complex dates from the Bronze Age, about 2000 BCE, and appears to have been in use for almost a millennium.
In 2010, the authors were able to state that:
"Our results demonstrated that the Xiaohe people were an admixture from populations originating from both the West and the East, implying that the Tarim Basin had been occupied by an admixed population since the early Bronze Age. To our knowledge, this is the earliest genetic evidence of an admixed population settled in the Tarim Basin."
20 individuals from the earliest layers of the cemetery complex were tested. This revealed that
"The dominant haplogroup, in the Xiaohe people was the East Eurasian lineage C"
which would correspond with an origin in Southern Siberia. However, the analysis also revealed that:
"two West Eurasian mtDNA haplogroups H and K were found among the Xiaohe people"
Obviously, this was of particular interest, and after further analysis, the authors concluded:
"Considering the presence of haplogroups H and K in the Xiaohe people and the geographical distribution of shared sequences, we conclude that the west Eurasian component observed in the Xiaohe people originated from western European, and maternal ancestry of the Xiaohe people might have close relationship with western European."
- [Chunxiang Li, et al, 2010]
The authors then analysed samples from later burial layers. Although the dominant haplogroup was again haplogroup C, the evidence:
"confirmed that the origin of the mitochondrial lineages is more widespread".
In all, the evidence to date records six west Eurasian lineages, five east Eurasian lineages, and one Indian lineage. As we already know, the Tarim Basin lay on the ancient Silk Road routes, so this genetic diversity is unsurprising.
- Map of Eurasia showing the location of the Xiaohe cemetery, the Tarim
Basin, the ancient Silk Road routes and the areas occupied by
cultures associated with the settlement of the Tarim Basin.
[Chunxiang Li, et al, 2015]
Overall, however, the authors conclude that the dominance of the:
"west Eurasian genetic components in the Xiaohe people corroborate the "steppe hypothesis"."
So, for now (in the absence of contradictory evidence from other sites), it seems that we can say with some confidence that the ancestors of the Tarim mummies originated in the steppes of southern Siberia.
Chunxiang Li, Hongjie Li, Yinqiu Cui, Chengzhi Xie, Dawei Cai, Wenying Li, Victor H Mair, Zhi Xu, Quanchao Zhang, Idelisi Abuduresule, Li Jin, Hong Zhu and Hui Zhou: Evidence that a West-East admixed population lived in the Tarim Basin as early as the early Bronze Age, BMC Biology, 2010, 8:15
Chunxiang Li, Chao Ning, Erika Hagelberg, Hongjie Li, Yongbin Zhao, Wenying Li, Idelisi Abuduresule, Hong Zhu and Hui Zhou: Analysis of ancient human mitochondrial DNA from the Xiaohe cemetery: insights into prehistoric population movements in the Tarim Basin, China, BMC Genetics, 2015, 16:78