  Let $\mathcal{A}$ be a unital $C^{*}$-algebra. We consider Jordan
$*$-homomorphisms on $C(X, \mathcal{A})$ and Jordan $*$-homomorphisms on
$\operatorname{Lip}(X,\mathcal{A})$. More precisely, for any unital
$C^{*}$-algebra $\mathcal{A}$, we prove that every Jordan $*$-homomorphism on
$C(X, \mathcal{A})$ and every Jordan $*$-homomorphism on
$\operatorname{Lip}(X,\mathcal{A})$ is represented as a weighted composition
operator by using the irreducible representations of $\mathcal{A}$. In
addition, when $\mathcal{A}_1$ and $\mathcal{A}_2$ are primitive
$C^{*}$-algebras, we characterize the Jordan $*$-isomorphisms. These results
unify and enrich previous works on algebra $*$-homomorphisms on $C(X,
\mathcal{A})$ and $\operatorname{Lip}(X,\mathcal{A})$ for several concrete
examples of $\mathcal{A}$.
