  In this article we study motives corresponding to the moduli stacks of
G-shtukas and their local models. In particular we deal with the question of
describing their motivic fundamental invariants. As an application, we provide
a criterion for mixed Tateness of the local model, and discuss the
semi-simplicity of Frobenius on their cohomology. We then use the theory of
local models to reformulate a purity result for these moduli stacks in the
motivic context.
