If we're talking currents in the outer core, then the inner core and the lower mantle are the surrounding material. Yes, it's liquid moving against those solid surfaces (which there is no reason to assume are rotating in sync with each other if the layer in between isn't).
I'm thinking more about this. I just can't see a sudden change from solid to liquid and back to solid within the Earth. I'm thinking it's more likely that it's a gradual change, that the viscosity decreases with temperature the deeper you go, until you reach a point where the pressure is so great that it begins to increase again until it's in effect a solid fluid. I think if you had a graph that shows the viscosity of the earth as you get deeper, you'd have a bell curve. Where that curve is at its lowest is where the flow is happening at its greatest, under the influence of gravity and the Earth's own rotation. The rotation keeps things flowing, while the gravity keeps things fluctuating.

I'm so glad that I haven't got a scientific background, otherwise I'd know how much shit I'm talking.