If you bring together two flat slabs of different materials at different starting temperatures, and only heat conduction is involved, then the interface between the two materials will almost instantly come to a temperature intermediate between the bulk temperatures of the two materials (and remain at that temperature for a long period). The interface temperature will depend on the thermal conductivities, the heat capacities, and the densities of the materials. Something similar happens when you touch an object with your hand, although conduction will not be the only mode of heat transfer on your side of the interface. See Carslaw and Jaeger, Conduction of Heat in Solids for the solution to the conductive slab problem.
Chet