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Diffusion models have shown remarkable potential in planning and control tasks due to their ability to represent multimodal distributions over actions and trajectories. However, ensuring safety under constraints remains a critical challenge for diffusion models. This paper proposes Constrained Diffusers, an extended framework for planning and control that incorporates distribution-level constraints into pre-trained diffusion models without retraining or architectural modifications. Inspired by constrained optimization, we apply a constrained Langevin samplingfor the reverse diffusion process that jointly optimizes the trajectory and realizes constraint satisfaction through three iterative algorithms: projected method, primal-dual method and augmented Lagrangian methods. In addition, we incorporate discrete control barrier functions as constraints for constrained diffusers to guarantee safety in online implementation, following a receding-horizon control that we generate a short-horizon plan and execute only the first action before replanning. Experiments in Maze2D, locomotion, and PyBullet ball running tasks demonstrate that our proposed methods achieve constraint satisfaction with less computation time, and are competitive with existing methods in environments with static and time-varying constraints.