The transient plane source technique has been used for simultaneous measurements of thermal conductivity and diffusivity in the lithium-sodium sulphate system starting near 300 K. Samples with 100, 77.5, or 50% Li2SO4 behave differently from pure Na2SO4. In the first case both the thermal conductivity and the diffusivity start to increase rapidly at about 640 K, while the crystal structure remains monoclinic for Li2SO4 and trigonal for LiNaSO4. Concerning pure Li2SO4, there is an additional discontinuous increase of the two thermal parameters at the structural transition from monoclinic to cubic (fcc) at 850 K, while the temperature gradients become significantly smaller in the cubic phase than in the monoclinic one. In contrast, both the thermal conductivity and the diffusivity of Na2SO4 decrease over the whole studied temperature range, which includes the phase transitions at 474 K and 520 K. Furthermore, a corresponding study was performed for silver iodide over the range 295-640 K, i.e, on both sides of the phase transition at 420 K. There is an indication of a small decrease of the thermal conductivity and the diffusivity at the phase transition. The high temperature phases fee Li2SO4, bcc LiNaSO4 and bcc AgI are solid electrolytes, but it is characteristic for the two sulphate phases that a coupled rotation-like motion of the sulphate ions enhances the cation migration. Obviously, such motion is also of importance for heat transport.