Context. Star-forming galaxies emit non-thermal radiation from radio to gamma rays. Observations show that their radio and gamma-ray luminosities scale with their star formation rates, supporting the hypothesis that non-thermal radiation is emitted by cosmic rays produced by their stellar populations. However, the nature of the main cosmic-ray transport processes that shape the emission in these galaxies is still poorly understood, especially at low star formation rates.

Aims. Our aim is to investigate the main mechanisms of global cosmic-ray transport and cooling in star-forming galaxies. The way they contribute to shaping the relations between non-thermal luminosities and star formation rates could shed light onto their nature, and allow us to quantify their relative importance at different star formation rates.

Methods. We developed a model to compute the cosmic-ray populations of star-forming galaxies, taking into account their production, transport, and cooling. The model is parametrised only through global galaxy properties, and describes the non-thermal emission in radio (at 1.4 GHz and 150 MHz) and gamma rays (in the 0.1 100 GeV band). We focused on the role of diffusive and advective transport by galactic winds, either driven by turbulent or thermal instabilities. We compared model predictions to observations, for which we compiled a homogeneous set of luminosities in these radio bands, and updated those available in gamma rays.

Results. Our model reproduces reasonably well the observed relations between the gamma-ray or 1.4 GHz radio luminosities and the star formation rate, assuming a single power-law scaling of the magnetic field (with index beta = 0:3) and winds blowing either at Alfvenic speeds (similar to tens of km s(-1), for less than or similar to 5 M-circle dot yr(-1)) or typical starburst wind velocities (similar to hundreds of km s(-1), for greater than or similar to 5 M-circle dot yr(-1)). Escape of cosmic rays is negligible for greater than or similar to 30 M-circle dot yr(-1). A constant ionisation fraction of the interstellar medium fails to reproduce the 150MHz radio luminosity throughout the whole star formation rate range.

Conclusions. Our results reinforce the idea that galaxies with high star formation rates are cosmic-ray 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 cosmic-ray or thermally driven at low and intermediate star formation rates, respectively. Our results globally support that magneto-hydrodynamic turbulence is responsible for the dependence of the magnetic field strength on the star formation rate and that the ionisation fraction is strongly disfavoured to be constant throughout the whole range of star formation rates.