When a person holding the cup of tea is descending on a spiral staircase, they're moving downwards and are rotating about its axis. Linear motion is irrelevant in this case, only the rotational part matters. And exactly the same thing happens as if you were to rotate the cup: it rotates relative to fluid inside. As boundary layer of the fluid that contacts the cup is dragged along with the cup, inner parts of the fluid are still stationary due to inertia, and this velocity gradient creates turbulence that can mix fluid inside if it has enough intensity.
In practice, however, I think it'd take quite a long staircase to mix a cup of coffee properly.