USA just posted a story on NASA's lasso mission ( link
) which has a video that's much more informative than the text.
It looks like they're using an expandable bucket for the capture phase (perhaps inflating like an air mattress, but I'm not exactly sure), and then they collapse it tightly around the asteroid. That part looked kind of like sucking air out of bag (which of course doesn't work in a vacuum). It also looked like they were about to start pulling it as you suggest, but they pushed it instead.
That's concept art recycled from Planetary Resources LLC, from whom NASA seems to have borrowed the idea. Planetary Resources has spent the last year studying different methods of capturing small asteroids in mission profiles more or less identical to what NASA describes, but they don't have anything like a working model for how that would really work. The concept art is just that.
Anyway, I recently red that some small chondrites are something between 30% and 50% void space internally, just a loose assemblage of boulders that happen to cling together. That might make any attempt at anchoring both easier (just stick a long pole in a hole or gap) and harder (It might just pull a couple rocks loose). One the other hand it might make processing easier because the material would already be in more manageable chunks.
More to the point, if you have the ability to break the entire asteroid into smaller chunks, you could simply wrap a cargo net around the whole thing and arrange them into a long chain and haul that along its own center of gravity.