In robotic deformable object manipulation (DOM) applications, constraints arise commonly from environments and task-specific requirements. Enabling DOM with constraints is, therefore, crucial for its deployment in practice. However, dealing with constraints turns out to be challenging due to many inherent factors, such as inaccessible deformation models of deformable objects (DOs) and varying environmental setups. This article presents a systematic manipulation framework for DOM subject to constraints by proposing a novel path set planning and tracking scheme. First, constrained DOM tasks are formulated into a versatile optimization formalism, which enables dynamic constraint imposition. Because of the lack of the local optimization objective and high state dimensionality, the formulated problem is not analytically solvable. To address this, planning of the path set, which collects paths of DO feedback points, is proposed subsequently to offer feasible path and motion references for the DO in constrained setups. Both theoretical analyses and computationally efficient algorithmic implementation of path set planning are discussed. Lastly, a control architecture combining path set tracking and constraint handling is designed for task execution. The effectiveness of our methods is validated in a variety of DOM tasks with constrained experimental settings.