We extend our recent study of diffusion in strongly chaotic systems (`the random model') to the general systems of mixed-type dynamics, including especially KAM systems, regarding the diffusion in chaotic components. We do this by introducing a Poissonian model as in our previous random model describing the strongly chaotic systems, except that now we allow for different a priori probabilities in different cells of the discretized phase space (surface of section). Thus the concept of greyness (of cells), denoted by g, such that
, is introduced, as is its distribution w(g). We derive the relationship between the dynamical property, namely the (normalized) fraction of chaotic component
as a function of discrete time j, and w(g). We predict again the universal scaling law, namely that for any w(g), the chaotic fraction
is a function of
only, and not separately of j and N, where N is the number of cells of equal size 1/N. The random model of exponential
is reproduced if all cells have g = 1, i.e.
. We argue that in two-dimensional systems, at any finite N, w(g) is non-trivial due to the fractal dimension of the boundary of the chaotic component, but is such that it goes to
as
, whilst in systems with three or more degrees of freedom w(g) has a well defined limit with non-zero values also at g<1. This is due to the existence of the Arnold web. We suggest how - through our formalism - one can calculate the Lebesgue measure of the chaotic component at each finite discretization, whose limit exists for
. Our findings are verified and illustrated for two- and three-dimensional billiards.