To maximize the performance of a distributed heterogeneous computing system, it is essential to assign resources to tasks (match) and order the execution of tasks on each resource (schedule) in a manner that exploits the heterogeneity of the resources and tasks. The mapping (defined as matching and scheduling) of tasks onto machines with varied computational capabilities has been shown, in general, to be an NP-complete problem. Therefore, heuristic techniques to find a near-optimal solution to this mapping problem are required. Dynamic mapping is performed when the arrival of a task is not known a priori and there may be changes in the system. In the heterogeneous environment considered in this study, tasks arrive randomly, tasks are independent (i.e., no communication among tasks), and tasks have priorities and multiple deadlines. This research proposes, evaluates, and compares eight dynamic heuristics.