Plant-based meat analogues that mimic the characteristic structure and texture of meat are becoming increasingly popular. They can be produced by means of high moisture extrusion (HME), in which protein-rich raw materials are subjected to thermomechanical stresses in the extruder at high water content (>40%) and then forced through a cooling die. The cooling die, or generally the die section, is known to have a large influence on the products’ anisotropic structures, which are determined by the morphology of the underlying multi-phase system. However, the morphology development in the process and its relationship with the flow characteristics are not yet well understood and, therefore, investigated in this work. The results show that the underlying multi-phase system is already present in the screw section of the extruder. The morphology development mainly takes place in the tapered transition zone and the non-cooled zone, while the cooled zone only has a minor influence. The cross-sectional contraction and the cooling generate elongational flows and tensile stresses in the die section, whereas the highest tensile stresses are generated in the transition zone and are assumed to be the main factor for structure formation. Cooling also has an influence on the velocity gradients and, therefore, the shear stresses; the highest shear stresses are generated towards the die exit. The results further show that morphology development in the die section is mainly governed by deformation and orientation, while the breakup of phases appears to play a minor role. The size of the dispersed phase, i.e., size of individual particles, is presumably determined in the screw section and then stays the same over the die length. Overall, this study reveals that morphology development and flow characteristics need to be understood and controlled for a successful product design in HME, which, in turn, could be achieved by a targeted design of the extruders die section.