We present a model-independent and relativistic approach to analytically derive electromagnetic finite-size effects beyond the point-like approximation. The key element is the use of electromagnetic Ward identities to constrain vertex functions, and structure-dependence appears via physical form-factors and their derivatives. We apply our general method to study the leading finite-size structure-dependence in the pseudoscalar mass (at order $1/L^3$) as well as in the leptonic decay amplitudes of pions and kaons (at order $1/L^2$). Knowledge of the latter is essential for Standard Model precision tests in the flavour physics sector from lattice simulations.