Sampling from discrete distributions is a ubiquitous task in machine learning, recently revisited by the emergence of discrete diffusion models. While Langevin algorithms constitute the state of the art for continuous spaces, discrete versions lack similar theoretical guarantees when the step-size becomes small. In this paper, we address this limitation by interpreting discrete sampling algorithms as discretizations of continuous-time dynamics on the hypercube. In particular, we describe several score functions for discrete algorithms which result in approximations of Glauber dynamics for the correct target distribution. We also compute upper bounds for the contraction of these algorithms, with or without Metropolis adjustment.