We study stochastic processes on combinatorial state spaces with local transition constraints, as arise in local search algorithms. We show that asymmetry in local transitions induces a systematic drift in a distance process relative to a reference configuration. This drift results from the imbalance between inward and outward transitions, translating combinatorial multiplicities into directional bias. Analyzing the random walk on the Johnson graph, we derive explicit expressions for the drift and expected hitting times. We also show that locality constraints lead to trajectory-level differences that can hinder search trajectories from reaching the target, even under identical stationary distributions.