A very important subproblem in the task assignment problem for unmanned air vehicles is the evaluation of costs for the state transitions of a directed graph. Usually a Dubins vehicle flying in the absence of wind is considered in the computation of costs. However, when a prevailing wind vector field is considered, the costs taken on very different values and the task assignment problem can have very different solutions. In this paper, we consider the problem of constructing minimum-time trajectories for a Dubins vehicle in the presence of a time-varying wind vector field. We present results on the existence and uniqueness of minimum-time solutions for a Dubins vehicle flying in a general time-varying wind vector field under some technical conditions. These results extend the conclusions of the well-known Dubins theorem. We also propose an algorithm for obtaining the minimum-time solution for an unmanned air vehicle and prove its convergence. We also present the results of numerical experiments that show that the importance of considering wind vector fields while planning the tour for an unmanned air vehicle.