Let $G$ be a unitary similitude group over $\mathbb{Q}$ (as in the book of Harris-Taylor), $\pi$ an irreducible automorphic representation of $G(\mathbb{A})$. I'm looking for some results on its multiplicity $m(\pi)$ in the space of automorphic forms. Do we know its exact value under certain conditions? e.g. suppose $\pi$ admit a cuspidal base change (in the sense of Cor.VI.2.8 of Harris-Taylor)?

Many thanks.