  We investigate the three types of class B Bianchi cosmologies filled with a
tilted perfect fluid undergoing velocity diffusion in a scalar field
background. We consider the two most importantcases: dust and radiation. A
complete numerical integration of the Einstein field equations coupled with the
diffusion equations is done to demonstrate how the presence of diffusion can
affect the dynamics of cosmological evolution, where the most attention is paid
to changes to the late-time behaviour. We show that aside from quantitative
effects, diffusion can result in significant qualitative differences. For
example, the cosmologies may recollapse if diffusion is sufficiently strong, or
evolve towards the de Sitter state otherwise. In constrast to the diffusionless
case, radiation isotropizes in presence of diffusion, and the tilt decreases
exponentially at later times: $V\sim e^{-0.25\tau}$; also, we determine the
decay rates of energy density, which become slower when the diffusion term is
non-zero.
