In this talk we address two important issues which could affect reaching the exponential and Kasner asymptotes in Einstein-Gauss-Bonnet cosmologies -- spatial curvature and anisotropy in
both three- and extra-dimensional subspaces. In the first part we consider cosmological evolution of spaces being the product of two isotropic and spatially curved subspaces. We consider all possible
number of spatial dimensions and provide description of the curvature effects in these dimensions. It is demonstrated that the dynamics in $D=2$ (the number of extra dimensions) and $D \ge 3$ is different.
In particular, the regime with the ``stabilization'' of extra dimensions could be reached only if $D \ge 3$.
In the second part we study the influence of initial anisotropy. Our study of reveals that transition from Gauss-Bonnet Kasner
regime to anisotropic exponential expansion (with expanding
three and contracting extra dimensions) is stable with respect to breaking the symmetry within both three- and extra-dimensional subspaces in any number of extra dimensions. This allows us to construct a scenario where isotropisation of outer and inner
subspaces is reached dynamically from rather general anisotropic initial conditions.