The effect of alloying on the structural and thermal properties of Cun-xAux (n=13,14) clusters is investigated by constant energy Molecular Dynamics simulations. The interactions between the atoms in the clusters are mimicked by a many-body (Gupta-like) potential based on the second moment approximation to the tight-binding model. The minimum energy structures and the lowest-lying isomers of the pure and mixed clusters are obtained by thermal quenching. We find icosahedral-like ground state structures for the 13- and 14-atom clusters and for all the concentrations, the only exception being Au-14 which has C-6v symmetry. Mixed structures are preferred over the segregated ones. The lowest-lying isomers of the binary clusters are the permutational ones; i.e., isomers having the same underlying geometry as the ground state structure and different relative arrangement of the unlike atoms in the atomic positions of the geometry. However, presence of these low lying permutational isomers does not affect the gross features of the melting-like transition. The 13- and 14-atom (icosahedral-like) binary clusters melt in one and two stages, respectively, as the corresponding pure Cu clusters. In constrast the melting-like transition of Au-14 exhibits a single stage. The melting temperature is studied as a function of cluster concentration and size. The main conclusion is that mixed Cu-Au clusters likely behave as pure Cu clusters both from the structural and the dynamical points of view, for all concentrations.