In fields ranging from fluid dynamics to optics, rich dynamic behavior of self-reinforcing solitary wave packets has attracted a great deal of interest among physicists and mathematicians alike. These solitons maintain their spatially-localized shape while propagating and typically emerge from a delicate balance of nonlinear and dispersive effects in the physical host medium. Solitons of a very different type, often called “topological solitons”, are topologically-nontrivial, spatially-localized nonsingular field configurations that are rarely associated with out-of-equilibrium dynamics, but rather are studied as static field configurations embedded in a uniform background. Their topologically-nontrivial configurations can be classified using homotopy theory, though their stability in real physical systems usually also requires nonlinearities. I will discuss how we realize reconfigurable motion of various chirality-stabilized topologically-nontrivial skyrmionic and knotted field configurations in chiral nematic LCs. Topological solitons exhibit directional motion both as individual objects and collectively, often spontaneously selecting and synchronizing their motion directions as this out-of-equilibrium process progresses. This motion is not accompanied with annihilation and generation of defects, can persist for months, and its direction can be controllably reversed. By using a combination of optical microscopy and 3D modeling of both the equilibrium free-energy-minimizing director structures and their temporal evolution, we uncover the physical mechanisms behind the soliton motion. I will demonstrate that this motion emerges mainly from spatially-asymmetric changes of director structures that evolve non-reciprocally upon the application and removal of an electric field, so that the periodic modulation of an applied field yields net translational motion of solitons.