This talk is concerned with Schur-constant survival models for discrete random variables. Our main purpose is to prove that the associated partial sum process is a non- homogeneous Markov chain. This is shown in different cases as the random variables take values in the set of nonnegative integers or in the set of integers smaller than $m\geq 1$. The property of Schur-constancy is also compared for thesecases. We also present a few additional results on Schur-constant vectors. This is based on joint works with Castaner, Claramunt, Lefèvre and Utev.