Ok, here's the real answer.
Yes they do work when the cop car is moving. The cop car is equiped with esentially two radars. The first radar shoots the target vehicle and obtains it's speed. Now obviously this isn't going to be accurate because if you're closing on the cop it's gonna be higher and if you're pulling away from the cop it'll read lower. This is where the second radar comes into play, it shoots the ground to the front of the cop car and obtains the cops cars speed. The radar unit will read out both the target vehicles closing/pulling away speed and the cop cars radar indicated speed, which the cop must check against his own speedometer. Then the radar unit calculates the figures and displays the target vehicle speed.
For example: You're driving 55 and closing on a cop who's driving 45. The radar would read your speed at 100. But the radar would take the cops speed of 45, and subtract it from the target speed of 100....to get your speed.