The mathematical foundations of the multipoint method are illustrated and the method is developed for the neutron kinetics of multiplying systems to treat physical situations in which spatial and spectral effects can play an important role in transient conditions, and hence the classical point-kinetic model can become inadequate. In the present paper the method is specifically developed for source-driven systems, through a proper adaptation of the factorization-projection technique used to derive other classic kinetic models. The results presented for some test cases show the advantages that can be attained with respect to the standard point model, even when treating relevant spatial and spectral transients. It is then shown how the technique can be inserted into a quasi-static framework.