This is a special case of restricting automorphic forms on a larger group $G$ to a smaller (sub)group $H$, of course. So far as I know, orthogonal groups are not special in this regard, although, yes, there are some obvious natural maps among them.

Certainly if the map $H\to G$ is a $k$-morphism (with groundfield $k$, or, being more careful, over some localization of the ring of integers of $k$), then the restrictions of left $G_k$-invariant functions are left $H_k$-invariant. This is the immediate part.

Under very mild hypotheses and/or normalizations, "right $K$-finiteness" is preserved, as is "moderate growth".

But $\mathfrak z$-finiteness (where $\mathfrak z$ is the center of the universal enveloping algebra) is very rarely preserved. Likewise, and in parallel, if we require automorphic forms to generate irreducibles under right translation, this property will very rarely be preserved under restriction.

The rarity of this is already visible on $\mathbb R$ with the Laplacian: very rarely is the product of two $\Delta$-eigenfunctions $\Delta$-finite...

In the automorphic context, indeed, computing decomposition coefficients of such a restriction (or product) occasionally produces very interesting Euler products. Rankin-Selberg and Langlands-Shahidi et al are instances of this.

EDIT: still, yes, it is true in some rather special situations (see work of Kudla and Rallis on "first occurrence", for example, and "Howe conjectures"), that for suitable $H_1\times H_2\subset G$, for rather degenerate (e.g., "minimal") (automorphic and other) repns on $G$, restriction to $H_1\times H_2$ and projection to certain $\pi_1$-components on $H_1$ produces an irreducible on $H_2$. (Also see Segal-Shale-Weil...)

This is not quite what is happening in the literal "map to Fourier-Jacobi coefficients" story. There we have a two-step-nilpotent abelian radical, and the map is "integrate along the center of that unipotent radical". This is a $G$-hom, so preserves $\mathfrak z_G$ eigenvalues. But it does *not* immediately promise eigenfunction properties for the Levi component. Yes, holomorphy is preserved, for example. Is this more in the direction of what you're wanting?