No, neither the silicon atoms, nor the dopant atoms can move ... they simply make the crystal lattice "imperfect". I suppose an atom may move back and forth between two sites, especially with changes in temperature, but this is more a behavior of surface atoms which are weakly bound.
The effect of the dopant is to change the conductivity of the silicon; it also makes it N or P type: a donor of negative (electrons) or positive (holes) charges ... as with the NP diode.
In other words, it is a method of engineering the properties of the material: the dopant and the density both change the properties.