  Starburst galaxies are efficient $\gamma$-ray producers, because their high
supernova rates generate copious cosmic ray (CR) protons, and their high gas
densities act as thick targets off which these protons can produce neutral
pions and thence $\gamma$-rays. In this paper we present a first-principles
calculation of the mechanisms by which CRs propagate through such environments,
combining astrochemical models with analysis of turbulence in weakly ionised
plasma. We show that CRs cannot scatter off the strong large-scale turbulence
found in starbursts, because efficient ion-neutral damping prevents such
turbulence from cascading down to the scales of CR gyroradii. Instead, CRs
stream along field lines at a rate determined by the competition between
streaming instability and ion-neutral damping, leading to transport via a
process of field line random walk. This results in an effective diffusion
coefficient that is nearly energy-independent up to CR energies of $\sim 1$
TeV. We apply our computed diffusion coefficient to a simple model of CR escape
and loss, and show that the resulting $\gamma$-ray spectra are in good
agreement with the observed spectra of the starbursts NGC 253, M82, and Arp
220. In particular, our model reproduces these galaxies' relatively hard GeV
$\gamma$-ray spectra and softer TeV spectra without the need for any
fine-tuning of advective escape times or the shape of the CR injection
spectrum.
