  We derive a set of light-cone sum rules relating the hadronic form factors
relevant for $B\to K\pi\ell^+\ell^-$ decays to the $B$-meson light-cone
distribution amplitudes (LCDAs). We obtain the sum rule relations for all $B\to
K\pi$ form factors of (axial)vector and (pseudo)tensor $b\to s$ currents with a
$P$-wave $K\pi$ system. Our results reduce to the known light-cone sum rules
for $B\to K^*$ form factors in the limit of a single narrow-width resonance. We
update the operator-product expansion for the underlying correlation function
by including a more complete set of $B$-meson LCDAs with higher twists, and
produce numerical results for all $B\to K^*$ form factors in the narrow-width
limit. We then use the new sum rules to estimate the effect of a non-vanishing
$K^*$ width in $B\to K^*$ transitions, and find that this effect is universal
and increases the factorizable part of the rate of $B\to K^*X$ decays by a
factor of $20\%$. This effect, by itself, goes in the direction of increasing
the current tension in the differential $B\to K^*\mu\mu$ branching fractions.
We also discuss $B\to K\pi$ transitions outside the $K^*$ window, and explain
how measurements of $B\to K\pi\ell\ell$ observables above the $K^*$ region can
be used to further constrain the $B\to K^*$ form factors.
