  We apply the dynamical systems approach to ever-expanding Bianchi type VIII
cosmologies filled with a tilted $\gamma$-fluid undergoing velocity diffusion
on a scalar field. We determine the future attractors and investigate the
late-time behaviour of the models. We find that at late times the normalized
energy density $\Omega$ tends to zero, while the scalar potential $\Phi$
approaches 1 and dominates the evolution. Moreover, we demonstrate that in
presence of diffusion fluids with $\gamma<3/2$, which includes physically
important cases of dust $(\gamma=1)$ and radiation $(\gamma=4/3)$, are
asymptotically non-tilted; the velocity of the fluid with $\gamma=3/2$ tends to
a constant value $0<\bar{V}<1$; and stiffer fluids evolve towards a state of
extreme tilt. Finally, we show that diffusion significantly reduces the decay
rates of energy density for dust and fluids stiffer than dust $(\gamma \geq
1)$; for example, at $\gamma=4/3$ (radiation) we obtain $\rho/H^2 \propto
e^{-3H_0 t}$ at late times, while $\rho/H^2 \propto e^{-4H_0 t}$ when diffusion
is absent.
