This paper presents the design and development of a magnetic-actuated multi-segment robotic catheter (MMR-C), which is driven by an external permanent mobile magnet system (EPMM). The structure of the multi-segment catheter is devised by the integration of variable outer diameter of each segment and multiple opposite polarization magnets. The system is designed for the target intervention application scenario of minimally invasive surgery in a multi-branch vascular cavity. The kinematic model of MMRC is derived based on the Cosserat rod method. The control scheme for the EPMM and the MMRC is introduced. The trajectories of the EPMM in Cartesian space are realized by the combination of dynamic movement primitives and Gaussian Mixture Regression. More over, the effectiveness of the proposed robotic system has been verified by conducting several experimental studies. The system performance is demonstrated by the carried out ring steering test and in-vitro vascular phantom intervention test.