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In this paper, we consider the dynamic multi-robot distribution problem where a heterogeneous group of networked robots is tasked to spread out and simultaneously move towards multiple moving task areas while maintaining connectivity. The heterogeneity of the system is characterized by various categories of units and each robot carries different numbers of units per category representing heterogeneous capabilities. Every task area with different importance demands a total number of units contributed by all of the robots within its area. Moreover, we assume the importance and the total number of units requested from each task area is initially unknown. The robots need first to explore, i.e., reach those areas, and then be allocated to the tasks so to fulfill the requirements. The multi-robot distribution problem is formulated as designing controllers to distribute the robots that maximize the overall task fulfillment while minimizing the traveling costs in presence of connectivity constraints. We propose a novel connectivity-aware multi-robot redistribution approach that accounts for dynamic task allocation and connectivity maintenance for a heterogeneous robot team. Such an approach could generate sub-optimal robot controllers so that the amount of total unfulfilled requirements of the tasks weighted by their importance is minimized and robots stay connected at all times. Simulation and numerical results are provided to demonstrate the effectiveness of the proposed approaches.