Everything we observe (depending on distance from the source) is from the past; light (electromagnetic radiation including radio waves) sound, even our senses of touch and smell are perceived after a slight delay.
As for radio waves transmitted from a source moving closer or further away relative to the receiver, these are subject to the
Doppler effect. That's how radar and laser speed detection works.
Moving away from a broadcast the pitch of music would be lowered and the signal would be delayed more as distance increased but neither of these phenomena would be observed by the receiver due to the time and distance dilation of relativity.
A moving transmitter would however be a completely different story. In this case Doppler shift and the increasing delay in the arrival of the signal would be observed by the stationary receiver. A transmitter moving away at the speed of light would simply disappear from our observable universe.