A theoretical analysis of the equilibrium geometry and thermal behavior of the ternary Li13Na30Cs12 alkali nanoalloy is presented. The calculations are based on the orbital-free approach to density functional theory and the classical Newtonian equations to deal with the electronic and atomic subsystems, respectively. An onion-like polyicosahedral structure is found to have the lowest energy, with a core shell formed by Li atoms and an external (incomplete) surface shell formed by Cs atoms, the remaining Na atoms forming an intermediate shell. In a narrow range of 10 meV/atom above the ground-state energy, we identify several other isomers, with varying compositional and structural disorder, but all of them based on a polyicosahedral growing pattern. The most important result extracted from an analysis of thermal properties is that diffusion of Cs atoms at the surface starts at approximate to 140 K, which is 50 K above typical surface melting temperatures of homogeneous Cs clusters. Thus we conclude that alloying may be useful in enlarging the thermodynamic stability of solid surfaces of clusters beyond its homogeneous limit. As the chemical reactivity of a cluster is known to be highly structure dependent, this observation may be especially relevant to heterogeneous catalysis and related applications. We also analyze the dynamical melting behavior of one of the higher-energy isomers and compare it to that of the ground-state structure.