  We prove explicit rationality-results for Asai- $L$-functions,
$L^S(s,\Pi',{\rm As}^\pm)$, and Rankin-Selberg $L$-functions,
$L^S(s,\Pi\times\Pi')$, over arbitrary CM-fields $F$, relating critical values
to explicit powers of $(2\pi i)$. Besides determining the contribution of
archimedean zeta-integrals to our formulas as concrete powers of $(2\pi i)$, it
is one of the advantages of our approach, that it applies to very general
non-cuspidal isobaric automorphic representations $\Pi'$ of ${\rm GL}_n(\mathbb
A_F)$. As an application, this enables us to establish a certain algebraic
version of the Gan--Gross--Prasad conjecture, as refined by N.\ Harris, for
totally definite unitary groups. As another application we obtain a
generalization of a result of Harder--Raghuram on quotients of consecutive
critical values, proved by them for totally real fields, and achieved here for
arbitrary CM-fields $F$ and pairs $(\Pi,\Pi')$ of relative rank one.
