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Data movement latency when using on-chip accelerators in emerging heterogeneous architectures is a serious performance bottleneck. While hardware/software mechanisms such as peer-to-peer DMA between producer/consumer accelerators allow bypassing main memory and significantly reduce main memory contention, schedulers in both the hardware and software domains remain oblivious to their presence. Instead, most contemporary schedulers tend to be deadline-driven, with improved utilization and/or throughput serving as secondary or co-primary goals. This lack of focus on data communication will only worsen execution times as accelerator latencies reduce. In this paper, we present RELIEF (RElaxing Least-laxIty to Enable Forwarding), an online least laxity-driven accelerator scheduling policy that relieves memory pressure in accelerator-rich architectures via data movement-aware scheduling. RELIEF leverages laxity (time margin to a deadline) to opportunistically utilize available hardware data forwarding mechanisms while minimizing quality-of-service (QoS) degradation and unfairness. RELIEF achieves up to 50% more forwards compared to state-of- the-art policies, reducing main memory traffic and energy consumption by up to 32% and 18%, respectively. At the same time, RELIEF meets 14% more task deadlines on average and reduces worst-case deadline violation by 14%, highlighting QoS and fairness improvements.