We consider a stochastic, dynamic runway scheduling problem involving aircraft landings on a single runway. Sequencing decisions are made with knowledge of the estimated arrival times (ETAs) of all aircraft due to arrive at the airport, and these ETAs vary according to continuous-time stochastic processes. Time separations between consecutive runway landings are modeled via sequence dependent Erlang distributions and are affected by weather conditions, which also evolve continuously over time. The resulting multi-stage optimization problem is intractable using exact methods and we propose a novel simheuristic approach, based on the application of methods analogous to variable neighborhood search (VNS) in a high-dimensional stochastic environment. Our model is calibrated using flight tracking data for over 98,000 arrivals at Heathrow Airport. Results from numerical experiments indicate that our proposed simheuristic algorithm outperforms an alternative based on deterministic forecasts under a wide range of parameter values, with the largest benefits being seen when the underlying stochastic processes become more volatile and also when the on-time requirements of individual flights are given greater weight in the objective function.