Star-forming galaxies (SFGs) emit non-thermal radiation from radio to gamma-rays. We aim to investigate the main mechanisms of global CR transport and cooling in SFGs. The way they contribute in shaping the relations between non-thermal luminosities and SFR could shed light onto their nature. We develop a model to compute the CR populations of SFGs, taking into account their production, transport, and cooling. The model is parameterised only through global galaxy properties, and describes the non-thermal emission in both radio and gamma-rays. We focus on the role of diffusive and advective transport by galactic winds, either driven by turbulent or thermal instabilities. We compare model predictions to observations, for which we compile a homogeneous set of luminosities in these radio bands, and update those available in gamma-rays. Our model reproduces reasonably well the observed relations between the gamma-ray or 1.4 GHz radio luminosities and the SFR, assuming a single power-law scaling of the magnetic field with the latter with index beta=0.3, and winds blowing either at Alfvenic speeds or typical starburst wind velocities. Escape of CR is negligible for > 30 Mo/yr. A constant ionisation fraction of the interstellar medium fails to reproduce the 150 MHz radio luminosity throughout the whole SFR range. Our results reinforce the idea that galaxies with high SFR are CR calorimeters, and that the main mechanism driving proton escape is diffusion, whereas electron escape also proceeds via wind advection. They also suggest that these winds should be CR or thermally-driven at low and intermediate SFR, respectively. Our results globally support that magnetohydrodynamic turbulence is responsible for the dependence of the magnetic field strength on the SFR and that the ionisation fraction is strongly disfavoured to be constant throughout the whole SFR range.
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