Dynamical properties of the domain walls (DW's) in the light beams propagating in nonlinear optical fibers are considered. In the bimodal fiber, the DW, as it was recently demonstrated numerically, separates two domains with different circular polarizations. This DW is found here in an approxiate analytical form. Next, it is demonstrated that the fiber's twist gives rise to an effective force driving the DW. The corresponding equation of motion is derived by means of the momentum-balance analysis, which is a technically nontrivial problem in this context (in particular, an effective mass of the DW proves to be negative). Since the sign of the twist-induced driving force depends on the DW's polarity, the DW's with opposite polarities can collide, which leads to the formation of their stable bound state. This is a domain of a certain circular polarization squeezed between semi-infinite domains of another polarization. In the absence of the twist, the DW can be driven by the Raman effect, but in this case the sign of the force does not depend on the DW's polarity and the bound state is not possible. Finally, a similar problem is considered for the dual-core fiber (coupler). In this case, the DW is a dark soliton in one core in the presence of the homogeneous field in the mate core. The dark soliton is driven by a force induced by the coupling with the mate core. The bound state of two dark solitons also exists in this system. The effects considered may find applications, e.g., for the optical storage of information.