  We study the homotopy theory of a certain type of diagram categories whose
vertices are in variable categories with a functorial path, leading to a good
calculation of the homotopy category in terms of cofibrant objects. The theory
is applied to the category of mixed Hodge diagrams of differential graded
algebras. Using Sullivan's minimal models, we prove a multiplicative version of
Beilinson's Theorem on mixed Hodge complexes. As a consequence, we obtain
functoriality for the mixed Hodge structures on the rational homotopy type of
complex algebraic varieties. In this context, the mixed Hodge structures on
homotopy groups obtained by Morgan's theory follow from the derived functor of
the indecomposables of mixed Hodge diagrams.
