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Coordinate-transformation-inspired optical devices have been mostly examined in the continuous-wave regime: the performance of an invisibility cloak, which has been demonstrated for monochromatic excitation, %would inevitably is likely to deteriorate for short pulses. Here we investigate pulse dynamics of flexural waves propagating in transformed plates. We propose a practical realization of a waveshifter and a rotator for flexural waves based on the coordinate transformation method. Time-resolved measurements reveal how the waveshifter deviates a short pulse from its initial trajectory, with no reflection at the bend and no spatial and temporal distortion of the pulse. Extending our strategy to cylindrical coordinates, we design a wave rotator. We demonstrate experimentally how a pulsed plane wave is twisted inside the rotator, while its wavefront is recovered behind the rotator and the pulse shape is preserved, with no extra time delay. We propose the realization of the dynamical mirage effect, where an obstacle appears oriented in a deceptive direction.