1. to carry away, to sweep away (In the literal sense of violently pulling someone or something along. Think of a flood that sweeps away a bunch of cars.)
2. to thrill, to captivate, to engage (Same idea as the first one, but now in a figurative sense of something being so engaging that it sweeps you with it. )
3. to tear "along" (Technically, the verb can also mean that something is tearing alng with something else. But it's hard to find an actual context for that. )