The manufacturing of large-scale structural components is still dominated by manual labor in many sectors of the modern composite industry. Efforts are being made to establish an automated layup technology for complex structural elements. Processing dry non-crimp fiber fabrics (NCF) offers great cost opportunities and high deposition rates, compared to prepreg-based technologies like automated fiber placement (AFP). Here, the fabric architecture is considered during the draping of the plane textile on curved surfaces. In this paper, the development of a draping unit for balancing fabric tension and consolidating continuously across the layup width is presented. We introduce a geometrical process model to achieve a fabric-friendly draping of the used unidirectional NCF. The shape of the resulting draping front depends on the surface geometry, the shearing of the previously laid-up textile, and the positioning of the material feed. To consolidate the fabric at the altering draping front in an automated layup process, the shape of the continuous consolidation element can be controlled by the elongation of serial soft actuators, manipulated by parallel robot kinematics. The shape replication ability of the draping unit is promising for the implementation of a continuous, fabric-friendly draping process for complex surface geometries.