Several nontrivial questions are discussed which arise in obtaining and utilizing boundary conditions along moving surfaces separating two media and which are associated with characteristics features of electrodynamic material equations. It is noted that the boundary conditions themselves can depend on the relationship between the thickness of the boundary and the proper times of the motion of the particles of the medium (and even simply on the velocity of the boundary). In view of the inertial properties of the medium all the electrodynamic quantities remain continuous (the law of continuity) at an ideally sharp discontinuity of its parameters in time. An exception is presented by the case of the motion of the boundary with velocity c, and also by discontinuities ``frozen into'' the medium. For a more smoothly varying (although sharply varying compared to the external scale of variation of the field) boundary layer one can neglect the inertial properties (dispersion) of the medium; in this case the field and the polarization undergo a ``discontinuity''. However, in this case difficulties of another kind arise when the velocity of the boundary is ``above light velocity'' on one side and ``below light velocity'' on the other side (in particular, the interaction of small perturbations with shock waves belongs to such a case); these cases require a separate investigation. The effect of the inertial properties of the medium on the boundary conditions is illustrated on the example of electromagnetic waves in a dielectric with elastic oscillators.