Generative models derived from large protein sequence alignments define complex fitness landscapes, but their utility for accurately modeling non-equilibrium evolutionary dynamics remains unclear. In this work, we perform a rigorous comparative analysis of three simulation schemes, designed to mimic evolution in silico by local sampling of the probability distribution defined by a generative model. We compare standard independent Markov Chain Monte Carlo, Monte Carlo on a phylogenetic tree, and a population genetics dynamics, benchmarking their outputs against deep sequencing data from four distinct in vitro evolution experiments. We find that standard Monte Carlo fails to reproduce the correct phylogenetic structure and generates gradual mutational sweeps. Performing Monte Carlo on a tree inferred from data adds phylogenetic fidelity and historical accuracy. The population genetics scheme successfully captures phylogenetic correlations, mutational abundances, and produces sweep-like dynamics as emergent properties, without the need to infer additional information from data. However, the latter choice comes at the price of not sampling the proper generative model distribution at long times. Our findings highlight the crucial role of phylogenetic correlations and finite-population effects in shaping evolutionary trajectories on fitness landscapes. These results clarify which features of laboratory evolution each scheme can capture, a prerequisite for using such models to explore evolutionary dynamics beyond current experimental limitations.