Loss of cuspidality by Langlands tranfer 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)?
 A: You are quite correct that the Langlands transfer map does not preserve cuspidality in general. E.g. if you take a modular form of CM type, coming from a Groessencharacter $\psi$ of some imaginary quadratic field K that doesn't factor through the norm map to $\mathbf{Q}$, then this gives you a cuspidal automorphic representation $\pi$ of $GL_2 / \mathbf{Q}$, but the base-change of $\pi$ to $GL_2 / K$ will be Eisenstein. 
This gives a flavour of what to expect: VERY roughly, the transfer of a cuspidal automorphic rep $\pi$ of $G$ to $G'$ will be cuspidal unless $\pi$ is itself a Langlands transfer from some other group $H$ via a homomorphism ${}^L H \to {}^L G$ whose image in ${}^L G'$ lands inside a Levi subgroup. 
This naive picture is, of course, far from being the whole story -- for instance, there are "CAP representations" which are cuspidal despite being lifts from Levi subgroups.
A lot of these subtleties were first seen in the case of $G = GSp_4$, $G' = GL_4$; this is where CAP forms were first identified, for instance. There is a nice survey article by Arthur from 2005. At the end of the article, he states a classification of discrete-spectrum automorphic reps of $GSp_4$ into six types A-F, of which type A ("general type") has cuspidal transfer to $GL_4$, and the other five types are all functorial liftings from various subgroups of $GSp_4$, whose transfers to $GL_4$ are easily seen to be Eisenstein. 
A lot of Arthur's work has been generalised to quasisplit unitary groups by Chung Pang Mok (Mem AMS, 2015) and that might go some way towards answering your more specific questions.
