Here we go, found it in the Health Impacts article:

There is experimental evidence that very slim fibers (<60 nm, <0.06 μm in breadth) tangle destructively with chromosomes (being of comparable size). This is likely to cause the sort of mitosis disruption expected in cancer.

And here in MECHANISMS OF ASBESTOS-INDUCED CARCINOGENESIS

It is somewhat more difficult to understand the “chromosome tangling hypothesis.” We recently found that asbestos fibers including crocidolite are actively taken up by several different kinds of cultured cells. Furthermore, those fibers enter both the cytoplasm and the nucleus. In this situation, asbestos fibers may tangle with chromosomes when cells divide. Whether there is a specificity of tangling for any chromosomal region is the next question to be addressed.

So not quite down to the DNA level, but basically chromosomes can get wrapped around asbestos fibers during cell division.