We derive a set of light-cone sum rules relating the hadronic form factors relevant for $Bto Kpiell^+ell^-$ decays to the $B$-meson light-cone distribution amplitudes (LCDAs). We obtain the sum rule relations for all $Bto Kpi$ form factors of (axial)vector and (pseudo)tensor $bto s$ currents with a $P$-wave $Kpi$ system. Our results reduce to the known light-cone sum rules for $Bto K^*$ 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 $Bto K^*$ 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 $Bto K^*$ transitions, and find that this effect is universal and increases the factorizable part of the rate of $Bto K^*X$ decays by a factor of $20%$. This effect, by itself, goes in the direction of increasing the current tension in the differential $Bto K^*mumu$ branching fractions. We also discuss $Bto Kpi$ transitions outside the $K^*$ window, and explain how measurements of $Bto Kpiellell$ observables above the $K^*$ region can be used to further constrain the $Bto K^*$ form factors.
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