Complex oxides with the general formula Pr1-xYxBaCo2-yNiyO6-δ (x = 0, 0.1, y = 0, 0.2) were successfully synthesized via combustion of organo-metallic precursors. The high-temperature dc conductivity of the obtained sintered materials was studied in a wide range of oxygen partial pressures and temperatures by means of the 4-probe method. The resulting dependencies were juxtaposed with the previously published data on oxygen non-stoichiometry for the oxides considered. The comprehensive analysis of these datasets in attempts to explain the observed trends has shown the large inadequacy of currently existing conduction models. Consequently, a new model approach was developed to account for the numerous experimental and theoretical peculiarities being characteristic for the cobaltites with a layered double perovskite structure. One of the key propositions made postulates mixed nature of the band structure for materials studied with spin states of Co ions acting as a spatial descriptor of a particular type of conductivity: semiconducting or a metallic one. The abovementioned hypothesis was validated by magnetic, thermodynamic and structural arguments obtained both theoretically and experimentally. The models suggested were shown to be adequate in describing large arrays of data collected. Additionally, the reasons behind doping and temperature/pressure influences on the respective conductivity changes in Pr1-xYxBaCo2-yNiyO6-δ were uncovered. Transport and thermodynamic parameters determined were used to evaluate transference numbers and mobilities of different charge carriers which revealed the dominating role of metallic conductivity under oxidative conditions and the superiority of semiconducting charge transport in reducing environments. The obtained conclusions were further supported by utilizing the derived model equations for successful description of conductivity/non-stoichiometry data for other layered cobaltites. Also, interesting correlations between cation composition, thermodynamic and transport properties were found. Finally, general review of the formulated approach was made and further research directions were proposed.