This paper introduces a new Dynamic Gated Recurrent Neural Network (DG-RNN) for compute-efficient speech enhancement models running on resource-constrained hardware platforms. It leverages the slow evolution characteristic of RNN hidden states over steps, and updates only a selected set of neurons at each step by adding a newly proposed select gate to the RNN model. This select gate allows the computation cost of the conventional RNN to be reduced during network inference. As a realization of the DG-RNN, we further propose the Dynamic Gated Recurrent Unit (D-GRU) which does not require additional parameters. Test results obtained from several state-of-the-art compute-efficient RNN-based speech enhancement architectures using the DNS challenge dataset, show that the D-GRU based model variants maintain similar speech intelligibility and quality metrics comparable to the baseline GRU based models even with an average 50% reduction in GRU computes.