A theoretical framework for the quasi-monochromatic electromagnetic (EM) processes such as excitation and propagation of long wave packets in dispersive, dissipative, bianisotropic media with weak and slow nonlinearity is developed. The time-dependent EM fields associated with such processes are expressed as products of two functions: the slowly varying complex amplitude (SVCA) and the quickly oscillating carrier. The material parameters are treated as operators acting on time-dependent EM fields. By expanding these operators in the Maxwell equations in a series with respect to a small time scale parameter a system of equations for the SVCAs of the EM fields in such media is formulated. In the linear case, the dynamic equations for the SVCAs that correspond to the transverse components of the electric and magnetic fields resemble the vector transmission line equations. The obtained system of equations is used to derive the dyadic Green functions for the SVCAs of the EM fields in bianisotropic media. This framework is applied for modeling propagation of partially coherent EM radiation in a material whose parameters may depend on the amount of the EM energy that has passed through it. The same framework can be used in studying propagation of modulated EM waves through a waveguiding system that includes bianisotropic metamaterial components. Applications of the developed formalism for the important special case of uniaxial bianisotropic media are outlined, with a few characteristic examples of beam and wave-packet propagation considered in detail.