We derive an analytical connection between kinetic relaxation rate and bulk viscosity of a relativistic fluid in d spatial dimensions, all the way from the ultra-relativistic down to the near non-relativistic regime. Our derivation is based on both Chapman–Enskog asymptotic expansion and Grad’s method of moments. We validate our theoretical results against a benchmark flow, providing further evidence of the correctness of the Chapman–Enskog approach; we define the range of validity of this approach and provide evidence of mounting departures at increasing Knudsen number. Finally, we present numerical simulations of transport processes in quark-gluon plasmas, with special focus on the effects of bulk viscosity which might prove amenable to future experimental verification. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.