  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.
