What's to stop some organisation from altering the orbit of an asteroid for some advantage? For example, an asteroid's orbit could be altered so that impacted on Chinese or American cities? I think there's going to need to be a lot of CCTV up in the big black so that such malice can be quickly detected and the consequences averted. Although such alterations could be used for constructive purposes also I guess. Perhaps comets could be redirected to land on the Moon or Mars to provide water.
If whoever gets there first is used, what's to stop someone chipping off a bit of the claimed asteroid and claiming it as theirs? How could it even be proven that a an asteroid was even once part of another, claimed one?
I think it might be better to instead have ownership of the mining vehicles and the material within them.
I suppose then you'd have big space vehicles that break apart and entirely envelop heliocentric materials for processing.