We develop a general method for calculating conduction-diffusion transport properties of strong electrolyte mixtures, including specific conductivities, steady-state electrophoretic mobilities, and self-diffusion coefficients. The ions are described as charged Brownian spheres, and the solvent-mediated hydrodynamic interactions (HIs) are also accounted for in the non-instantaneous ion atmosphere relaxation effect. A linear response expression relating long-time partial mobilities to associated dynamic structure factors is employed in our derivation of a general mode coupling theory (MCT) method for the conduction-diffusion properties. A simplified solution scheme for the MCT method is discussed. Analytic results are obtained for transport coefficients of pointlike ions which, for very low ion concentrations, reduce to the Deby-Falkenhagen-Onsager-Fuoss limiting law expressions. As an application, an unusual non-monotonic concentration dependence of the polyion electrophoretic mobility in a mixture of two binary electrolytes is discussed. In addition, leading-order extensions of the limiting law results are derived with HIs included. The present method complements a related MCT method by the authors for the electrolyte viscosity and shear relaxation function [C. Contreras-Aburto and G. Nägele, J. Phys.: Condens. Matter 24, 464108 (2012)], so that a unifying scheme for conduction-diffusion and viscoelastic properties is obtained. We present here the general framework of the method, illustrating its versatility for conditions where fully analytic results are obtainable. Numerical results for conduction-diffusion properties and the viscosity of concentrated electrolytes are presented in Paper II [C. Contreras Aburto and G. Nägele, J. Chem. Phys. 139, 134110 (2013)].