We provide a dispersion-theoretical representation of the reaction amplitudes $\gamma K\rightarrow K \pi$ 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.