Mixture-of-Experts (MoE) models offer immense capacity via sparsely gated expert subnetworks, yet adapting them to multiple domains without catastrophic forgetting remains an open challenge. Existing approaches either incur prohibitive computation, suffer cross-domain interference, or require separate runs per domain. We propose DES-MoE, a dynamic expert specialization framework for multi-domain adaptation of Mixture-of-Experts models. DES-MoE addresses catastrophic forgetting through three innovations: (1) an adaptive router balancing pre-trained knowledge retention and task-specific updates via distillation, (2) real-time expert-domain correlation mapping to isolate domain-specific gradients, and (3) a three-phase adaptive fine-tuning schedule that progressively freezes non-specialized parameters. Evaluated on six domains (math, code, law, etc.), DES-MoE matches single-domain ESFT performance while training one unified model, reduces forgetting by 89% compared to full fine-tuning as domains scale from 2 to 6, and achieves 68% faster convergence than conventional methods. Our work establishes dynamic expert isolation as a scalable paradigm for multi-task MoE adaptation.