Sparse Mixture of Experts (SMoE) architectures reduce computational cost by activating only a subset of experts per token, yet they often retain large memory footprints and exhibit significant redundancy, both within and across layers. We propose GMoE, a sparse MoE architecture designed to explicitly address these inefficiencies. Instead of maintaining separate expert sets for each layer, GMoE uses Global Experts shared across all layers and adds a Local Expert per layer for layer-specific adaptation. This architecture reuses Global Experts across layers, thereby mitigating inter-layer redundancy while substantially reducing model parameters. In addition, we introduce a Global Router with a GRU-based recurrent component shared across layers and layer-specific routing heads that propagate routing logits across layers. This routing mechanism couples routing decisions across layers, progressively refines routing paths, and helps mitigate intra-layer redundancy. Across diverse language modeling corpora and downstream benchmarks, GMoE remains competitive while using substantially fewer parameters. Routing path analyses and an ablation study show that GMoE reduces cross-layer routing concentration and increases path diversity, with the Global Experts, the Local Expert, and the Global Router all contributing to the gains. The code is available at https://github.com/GEONWOOHONG/GMoE.