Is there in the literature any explicit combinatorial description of the branching rules from $\operatorname{SL}(n+m)$ to $\operatorname{SL}(n) \times \operatorname{SL}(m)$, from $\operatorname{SO}(n+m)$ to $\operatorname{SO}(n) \times \operatorname{SO}(m)$, and/or from $\operatorname{Sp}(2n+2m)$ to $\operatorname{Sp}(2n) \times \operatorname{Sp}(2m)$?
To give an idea of what I mean by "explicit", for example, explicit descriptions of the branching rules from $\operatorname{SL}(n+1)$ to $\operatorname{SL}(n)$, from $\operatorname{SO}(n+1)$ to $\operatorname{SO}(n)$ and from $\operatorname{Sp}(2n+2)$ to $\operatorname{Sp}(2n)$ are well-known, due to Weyl, Murnaghan and possibly Zhelobenko. They are given for example on the Wikipedia page for "Restricted representation", or in Knapp's "Lie groups beyond an introduction" (2d edition) in section 9.3. So ideally, I would like each of the rules from $G(n+m)$ to $G(n) \times G(m)$ to be sufficiently explicit that the rule from $G(n+1)$ to $G(n)$ basically appears as the special case $m = 1$.
For the rules I am asking about, the only thing I have found so far is this paper by Howe, Tan and Willenbring. However they give the branching rules only for a subset of the possible highest weights (what they call the "stable range"). Is there a description of these branching rules that also works outside the stable range?
Also, in $\operatorname{SL}(n+m)$, the subgroup $\operatorname{SL}(n) \times \operatorname{SL}(m)$ is Levi, so we can use Littelmann's branching rules in terms of the path model (see e.g. on Wikipedia). However making this explicit is still a nontrivial problem of combinatorics. Even if it turns out to be an easy one, I would still rather cite its solution than have to write it up on my own.
More generally, is there anywhere a survey with a list of the branching rules for which combinatorial descriptions are known? I would expect such a list to be found, if not in the "official" published literature, then possibly on some website - e.g. the Atlas project (although it seems to me that they only care about infinite-dimensionnal representations - correct me if I am wrong) or somewhere in the Sage documentation (like in the vicinity of here).
