Doppler Shift and the Pulse Repetition Interval
Actually the PRI also changes. There is a complete explanation about this topic in "Cobbold - Foundations of Biomedical Ultrasound" chapter 10 (Pulsed Methods for Flow Velocity Estimation and Imaging).

Consider a case where PRI of emitted pulse is 1 s and the wave speed is 9 m/s and the object is 10 m away and moving with speed of 1 m/s toward the antenna. The first echo will be received 2 seconds after the emission (after 1 seconds, the object moved 1 meter toward the antenna and placed 9 meter away from the antenna and the transmitted wave traveled 9 meter away from the antenna). For the next pulse the echo will be received 1.8 seconds after the emission (considering the object is moved closer to the antenna during the PRI between first and second pulse). So the first echo received at t=2 and the second echo received at t=2.8 and the PRI would be 0.8 s (the same is true for the next pulses until the object passes over the antenna).

Also the transformation you have here is something like x(t - (x0-vt)/c ).