  We provide a dispersion-theoretical representation of the reaction amplitudes
$\gamma K\to K \pi$ in all charge channels, based on modern pion-kaon $P$-wave
phase shift input. Crossed-channel singularities are fixed from phenomenology
as far as possible. We demonstrate how the subtraction constants can be matched
to a low-energy theorem and radiative couplings of the $K^*(892)$ resonances,
thereby providing a model-independent framework for future analyses of
high-precision kaon Primakoff data.
