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Notes: This paper is devoted to a systematic investigation of linear transport properties in strongly coupled binary ionic mixtures of pointlike ions interacting solely through Coulomb interactions. The basic formalism rests upon suitable extensions of the Boltzmann–Ziman equation. Validity conditions for the Lorentzian approximation are thoroughly discussed. High temperature and inelastic contributions to electron transport are emphasized. The formalism is, hereafter, specialized to a thorough investigation of electric, thermal, and mechanical transport coefficients. Basic transport quantities are expressed under a reduced form that allows an easy analytical treatment of temperature and inelastic corrections, parametrized with α=T/TF and (large-closed-square)=β(h-dash-bar)ω, respectively. The former are derived from exact solutions of transport equation through various jellium dielectric functions. Calculation of inelastic contributions is performed through the variational method. Electron transport at T=0 is then thoroughly investigated, including electric and thermal conductivities as well as thermopower and shear viscosity. These results are furthermore extended to an exact calculation of the electric conductivity up to order α2 including properly inelastic contributions, derived in terms of successive moments of the ion–ion structure factors.Results are presented under an analytic and compact form, convenient for a numerical implementation to strongly coupled H+–He2+ mixtures of astrophysical interest. Calculations are performed within binary ionic mixture (BIM) and polarized BIM (PBIM) frameworks, respectively. Ion–ion structure factors are computed through the hypernetted-chain (HNC) scheme, by neglecting bridge diagrams. Most of these results pertain to homogeneous phases. First, the focus is on BIM and a systematic investigation of the Γ-rs and c2 dependence of the transport quantities is performed. Also, attention is paid to the influence of the jellium dielectric function. The T=0 elastic limit is considered first. Next, inelastic and finite temperature corrections are addressed, and BIM to PBIM results are compared. Finally, the electric resistivity behavior is investigated in the vicinity of critical demixing. Discrepancy between BIM and PBIM are thus stressed.