Given an $L$-homomorphism of Langlands dual groups $${}^LG \to {}^LG'$$

Langlands functoriality contectures predicts the existence of a tranfer map of automorphic representations $$Aut(G) \to Aut(G')$$

However, nothing in the functoriality results or conjectures seems to be concerned with cuspidality. Let us say I am interested in the set $Cusp(G)$ of cuspidal representations of $G$. The above map gives a a tranfer from it to $Aut(G')$.

My question is: **what do we know about its image?** Is it also cuspidal? Is it endowed at least with some extra properties?

Since those questions have in general negative answers I believe, I am more precisely interested in unitary groups: what about $G$ be a quasi-split unitary group in 2 or 3 variables, and $G'=GL_4$ or $GL_6$ on the quadratic extension defining the unitary group (from the base change transfer)?