MUSICA will orchestrate an MSC-Action where the deepening of our knowledge of the local Universe will play together with unveiling the cosmic dawn. We will focus on the origin of new structures, stains, in the diffuse Galactic interstellar medium (ISM) recently revealed by the LOFAR radio telescope and model their contamination to the observations of the Epoch of Reionization. The study of the diffuse ISM is key for pivotal astrophysical problems, such as the inefficiency of the star formation process. The ISM fills the Galaxy with cosmic rays, multiphase gas, and dust particles, all coupled with magnetic fields. It is through their interactions that a complex cycle, involving phase transitions, magneto-hydrodynamic (MHD) turbulence, and gravity, is expected to convert diffuse/warm matter into denser/colder regions, where stars form at a rate of only a few solar masses per year. In theory, magnetic fields in the ISM may explain such low star-formation rate. In practice, characterizing magnetism at the starting point of this cycle of matter has been a difficult observational challenge until now. Today, the Planck satellite and LOFAR are unique complementary datasets to investigate phase transition between warm/ionized and atomic magnetized ISM. MUSICA proposes an unprecedented joint analysis of the Planck/LOFAR data with the best-available kinematic data of the atomic and ionized hydrogen to investigate the MHD origin of the stains. By comparing observations with synthetic data from state-of-the-art MHD simulations of the ISM, MUSICA will (1) significantly advance our knowledge of structure formation in the diffuse ISM and (2) prepare foreground analyses for future cosmological probes, such as LOFAR and the SKA radio telescopes. Hence, funding MUSICA will allow partners to (i) start a scientific network across Europe (France/Croatia) in line with the H2020 priorities, (ii) do science on the front line and (iii) secure their research in novel international projects.