Modern and future services require ultra-reliable mobile connections with high bandwidth parameters and proper security protection. It is possible to ensure such conditions by provisioning services in the Multi-Access Edge Computing system integrated with fifth-generation mobile networks. However, the main challenge in the mentioned architecture is providing a secure service migration with all related data and security requirements to another edge computing host area when the user changes its physical location. This article aims to present the state of research on the migration of the security context between service instances in Edge/MEC servers, specify steps of the migration procedure, and identify new security challenges inspired by use cases of 5G vertical industries. For this purpose, the analysis of the security context’s structure and basic concept of the Security Service Level Agreement was performed and presented in the document. Next, a further investigation of the security context was conducted, including requirements for its reliable migration between edge serves instances. The study mainly focused on crucial migration challenges and possible solutions to resolve them. Finally, the authors presented how the proposed solution can be used to protect 5G vertical industries services based on several mobile use cases.