In some interesting cases, decomposing an automorphic form/representation on a bigger group $G$ along a smaller group $H$ produces Euler products and similar helpful outcomes. Perhaps not "in general", especially if the subgroup $H$ is "too small". Both Rankin-Selberg and Langlands-Shahidi -type presentations of $L$-functions can be viewed exactly as saying that the decomposition coefficients in such certain such situation are $L$-functions.

E.g., even in the very simplest circumstance, restricting an elliptic modular cuspform to the split torus and decomposing, the "decomposition coefficients" are the $L$-functions associated to the cuspform via Mellin transform.

The case $GL_{n-1}\subset GL_n$, spectrally decomposing a cuspform $f$ on the larger along the smaller, produces the more general analogue of the previous, namely, the Hecke-type Rankin-Selberg product $L(s,f\times F)$ where $F$ is a cuspform on the smaller.

In some cases, the decomposition of the restriction can be completely understood, as in the case of $Sp_m\times Sp_n \to Sp_{m+n}$ and restriction of Siegel-type Eisenstein series (e.g., see my Taniguchi/Katata Symposium paper, 1984).

The cuspidal components of the restriction to $SL_2\times SL_2\times SL_2\to Sp_3$ (or $Sp_6$, if one prefers) are triple-product $L$-functions.

But, yes, in general, one cannot expect to say much.