Traffic in optical backbone networks is evolving rapidly in terms of type, volume, and dynamicity following the rapid growth of cloud-based services, ongoing adoption of 5G communications, and explosion of Internet of Things (IoT). Elastic Optical Network (EON), by adopting a flexible grid, can provide the required capacity and flexibility to handle these rapid changes. However, operators rarely perform greenfield deployments, so to limit upfront investment, a gradual migration from fixed-grid to flexible-grid switching equipment is preferable. For gradual migration, switching nodes can be upgraded (starting from bottleneck network links) while keeping the rest of the traditional fixed-grid network operational. We refer to the co-existence of fixed-grid and flex-grid optical equipment as a "mixed-grid" network. Traditional algorithms for dynamic resource assignment in EON will not effectively be applicable in a mixedgrid network due to inter-operability issues among fixed and flex-grid nodes. In this study, we propose a new algorithm, called Mixed-grid-aware Dynamic Resource Allocation (MDRA), to solve the route, spectrum, and modulation-format allocation (RSMA) problem in a mixed-grid network while considering inter-operability constraints. Our numerical results (on representative network topologies) show that the proposed method achieves 41% less blocking (for 50% offered load) compared to traditional approach. The proposed method also can gain about 15% more spectrum utilization for same load.