Let $G$ be a reductive group and $\lambda$, $\mu$ and $\nu$ be dominant weights of $G$. Denote by $V_\lambda$ the irreducible representation of $G$ of highest weight $\lambda$. It seems to be true that $$ V_\lambda \otimes V_\mu^* \subseteq V_{\lambda+\nu}\otimes V_{\mu+\nu}^*, $$ where $W^*$ as usual is the dual representation for $W$. This is obvious in the case $\lambda=0$ or $\mu=0$. Here are two questions:

(1) about multiplicities, i.e. how to prove that for any $\theta$ the multiplicity of $V_\theta$ in right hand side is more or equal than in the left hand side?

(2) is there a natural inclusion?