As the network grows, the centralized SDN (Software-Defined Networking) architecture causes a scalability problem. To handle this problem, multiple controllers have been used in many related works. However, if the load is concentrated on a certain controller, incoming messages to the controller can be blocked while other controllers are not busy. Especially, this problem becomes critical for handover requests in mobile networks because users will perceive severe QoS deterioration when these requests are blocked. Therefore, this paper presents a mobility-aware load distribution scheme for scalable SDN-based mobile networks. In the proposed scheme, incoming handover request messages can be differentiated among other messages and admitted with priority. In addition, these messages can be migrated to other controllers when the controller has no capacity to prevent them from being blocked. Analytical results confirm that the proposed scheme has lower blocking probability and higher resource utilization ratio than conventional schemes without much additional signaling load as offered load increases.