This may be semantics, but the swash plate doesn't move the rotor disk, it changes the pitch in the rotor blades based on input received through the intermixing bell crank from the cyclic/collective. This change in pitch tilts the rotor disk.
That's pretty much what I said. The cyclic and collective sticks tilt (and elevate) the lower swashplate and the upper swashplate translates that orientation into the rotating reference frame of the rotor. The actual variation of blade pitch is controlled by pushrods and bellcranks attached to the upper swashplate, and the rotor then tilts into the swashplate's orientation.
In other words, the swashplate doesn't move the rotor disk, but tells it where to move itself.