The fully accomplished standardization of the new mobile generation has led to the deployment of fifth-generation (5G) wireless networks to gratify enormous traffic volume for Internet services. The current centralized mobility system could not be sufficient to manage an explosive increase in data volume and is considered a steadily rising issue in modern wireless communication. A new technique that can affluently handle traffic problems and completely avoid network breakdown chances is indispensable. Recently, distributed mobility management (DMM) was introduced to overcome the inevitable obstacles that destructively impact the existing networks. Specifically, a novel design based on the deployment of distributed mobility anchors, closer to the terminal points, was introduced. Several works have been proposed to build DMM solutions with different focuses for 5G-and-beyond networks (B5G), which are also referred to as sixth-generation solutions (6G). In this paper, we present the potential and benefits of flat network design for efficient and fast routing of traffic and furnish the effectiveness of the scheme toward mobility management in B5G by delineating recent research works. We also present the current limitations, challenges, and future research directions for seamless mobility to achieve the desired objectives in the current 5G and upcoming 6G cellular communications.