  We give an adelic version of a spectral reciprocity formula relating
$\mathrm{GL}_3 \times \mathrm{GL}_2$ with $\mathrm{GL}_3 \times \mathrm{GL}_1$
and $\mathrm{GL}_1$ moments of $L$-functions discovered by Xiaoqing Li. For
many types of the $\mathrm{GL}_3$ representation, we describe the local weight
transforms via a decomposition of Voronoi's formula in terms of elementary
transforms, which generalizes the one given by Miller--Schmid in a way
consistent with the local Langlands correspondence. We obtain some integral
representation of the Bessel functions for $\mathrm{GL}_2$ uniformly over all
local fields. We announce the dual weight computation in a new case, which
identifies the coset problem parallel to that in Petrow--Young's Weyl-type
subconvexity for general Dirichlet $L$-functions. The proof and extensive study
of other accessible cases will be given in a separate paper.
