  We revisit the theoretical predictions and the parametrization of non-local
matrix elements in rare $\bar{B}_{(s)}\to \lbrace
\bar{K}^{(*)},\phi\rbrace\ell^+\ell^-$ and $\bar{B}_{(s)}\to \lbrace
\bar{K}^{*}, \phi\rbrace \gamma$ decays. We improve upon the current state of
these matrix elements in two ways. First, we recalculate the hadronic matrix
elements needed at subleading power in the light-cone OPE using $B$-meson
light-cone sum rules. Our analytical results supersede those in the literature.
We discuss the origin of our improvements and provide numerical results for the
processes under consideration. Second, we derive the first dispersive bound on
the non-local matrix elements. It provides a parametric handle on the
truncation error in extrapolations of the matrix elements to large timelike
momentum transfer using the $z$ expansion. We illustrate the power of the
dispersive bound at the hand of a simple phenomenological application. As a
side result of our work, we also provide numerical results for the $B_s \to
\phi$ form factors from $B$-meson light-cone sum rules.
