Based on the Galilean relativity principle and Maxwell's equations, electromagnetic field equations are derived for inertial frames, in which the substratum of the electromagnetic waves flows with arbitrary velocity | w | < c (velocity of light). It is demonstrated that the electromagnetic field equations with electromagnetic substratum flow are strictly covariant against Galilei transformations. Wave equations, conservation and invariance theorems, and boundary conditions are derived for the electrodynamic fields in presence of electromagnetic substratum flow. Initial-boundary-value problems are solved for electromagnetic signal propagation and induction in the substratum by an integral equation method. Physical effects for the measurement of the velocity field of the electromagnetic substratum are discussed. Maxwell's conception that his equations refer to a frame of reference with resting electromagnetic substratum is confirmed, and it is shown that Maxwell's equations are also applicable to inertial frames with small substratum velocities, | w | ≪ c.