  We study some tempered endoscopic cases of Langlands functoriality on the
$n$-variable unitary groups via the simple stable trace formula. This extends
previous work of Rogawski and Clozel-Harris-Labesse. Ramakrishnan and
Kim-Shahidi have proved the existence of the automorphic tensor product for
$GL_2 \times GL_2$ and $GL_2 \times GL_3$. We apply our endoscopic results to,
in a sense, "lift" these theorems and deduce the existence of the automorphic
tensor product for the quasi-split unitary groups $U^*_2 \times U^*_2$ and
$U^*_2 \times U^*_3$. We then formulate the notion of overconvergent Langlands
functoriality on the definite unitary group. We apply some results of Chenevier
to obtain morphisms of eigenvarieties that extend classical endoscopic
Langlands functoriality. This enables us to obtain, in the \emph{small slope}
setting, some classical cases of non-tempered endoscopic Langlands
functoriality. To conclude, we apply the recent potential automorphy results of
Barnet-Lamb-Gee-Geraghty-Taylor to obtain some cases of the potential
automorphic tensor product for higher rank unitary groups.
