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We apply the dynamical systems approach to ever-expanding Bianchi type VIIIcosmologies filled with a tilted $\gamma$-fluid undergoing velocity diffusionon a scalar field. We determine the future attractors and investigate thelate-time behaviour of the models. We find that at late times the normalizedenergy density $\Omega$ tends to zero, while the scalar potential $\Phi$approaches 1 and dominates the evolution. Moreover, we demonstrate that inpresence of diffusion fluids with $\gamma<3/2$, which includes physicallyimportant cases of dust $(\gamma=1)$ and radiation $(\gamma=4/3)$, areasymptotically non-tilted; the velocity of the fluid with $\gamma=3/2$ tends toa constant value $0<\bar{V}<1$; and stiffer fluids evolve towards a state ofextreme tilt. Finally, we show that diffusion significantly reduces the decayrates of energy density for dust and fluids stiffer than dust $(\gamma \geq1)$; for example, at $\gamma=4/3$ (radiation) we obtain $\rho/H^2 \proptoe^{-3H_0 t}$ at late times, while $\rho/H^2 \propto e^{-4H_0 t}$ when diffusionis absent.

The late-time behaviour of tilted Bianchi type VIII universes in presence of diffusion