In this talk, we consider a simple random walk defined on a Chung-Lu directed graph, an inhomogeneous random network that extends the Erdos Renyi random digraph by including edges independently according to given Bernoulli laws. In this non-reversible setting, we will focus on the convergence toward the equilibrium of the dynamics. In particular, under the assumption that the average degree grows logarithmically in the size n of the graph (weakly dense regime), we will establish a cutoff phenomenon at the entropic time of order log(n)/loglog(n). We will then show that on a precise window the cutoff profile converges to the Gaussian tail function. This is qualitatively similar to what was proved in a series of works by Bordenave, Caputo, Salez for the directed configuration model, where degrees are deterministically fixed. In terms of statistical ensembles, this analysis provides an extension of these cutoff results from a hard to a soft-constrained model. Joint work with G. Passuello.