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

Exploring the physics behind the non-thermal emission from star-forming galaxies detected in gamma rays