We study theoretically electromagnetic radiation emitted by inelastic Cooper-pair tunneling. We consider a dc-voltage-biased superconducting transmission line terminated by a Josephson junction. We show that the generated continuous-mode electromagnetic field can be expressed as a function of the time-dependent current across the Josephson junction. The leading-order expansion in the tunneling coupling, similar to the P(E) theory, has previously been used to investigate the photon emission statistics in the limit of sequential (independent) Cooper-pair tunneling. By explicitly evaluating the system characteristics up to the fourth order in the tunneling coupling, we account for dynamics between consecutively tunneling Cooper pairs. Within this approach we investigate how temporal correlations in the charge transport can be seen in the first- and second-order coherences of the emitted microwave radiation.