  Let $G$ be a complex simple Lie group, and let $U \subseteq G$ be a maximal
compact subgroup. Assume that $G$ admits a homogenous space $X=G/Q=U/K$ which
is a compact Hermitian symmetric space. Let $\mathscr{L} \rightarrow X$ be the
ample line bundle which generates the Picard group of $X$. In this paper we
study the restrictions to $K$ of the family $(H^0(X, \mathscr{L}^k))_{k \in
\N}$ of irreducible $G$-representations. We describe explicitly the moment
polytopes for the moment maps $X \rightarrow \fk^*$ associated to positive
integer multiples of the Kostant-Kirillov symplectic form on $X$, and we use
these, together with an explicit characterization of the closed $K^\C$-orbits
on $X$, to find the decompositions of the spaces $H^0(X,\mathscr{L}^k)$. We
also construct a natural Okounkov body for $\mathscr{L}$ and the $K$-action,
and identify it with the smallest of the moment polytopes above. In particular,
the Okounkov body is a convex polytope. In fact, we even prove the stronger
property that the semigroup defining the Okounkov body is finitely generated.
