Raspberries propagate from root cuttings. Black/yellow raspberries and blackberries are tip-rooting. You propagate them by staking the tip of a primocane to some soil. It is possible to dig up a blackberry bush and divide it, but it is not as good as tip rooting. I have heard that root cuttings do not work as well with blackberries.
I have had success propagating blackberry by carefully digging up suckers with significant root mass attached. They tend to get a slow start.