In the dynamic distributed task assignment (DDTA) problem, a team of agents is required to accomplish a set of tasks while maximizing the overall team utility. An effective solution to this problem needs to address two closely related questions: first, how to find a near-optimal assignment from agents to tasks under resource constraints, and second, how to efficiently maintain the optimality of the assignment over time. We address the first problem by extending an existing forward/reverse auction algorithm which was designed for bipartite maximal matching to find an initial near-optimal assignment. A difficulty with such an assignment is that the dynamicity of the environment compromises the optimality of the initial solution. We address the dynamicity problem by using swapping to locally move agents between tasks. By linking these local swaps, the current assignment is morphed into one which is closer to what would have been obtained if we had re-executed the computationally more expensive auction algorithm. In this paper, we detail the application of this dynamic auctioning scheme in the context of a UAV (Unmanned Aerial Vehicle) search and rescue mission and present early experimentations using physical agents to show the feasibility of the proposed approach.