Delay? Well for starters there is the transcoding delay from the A->D and D->A conversions, then propagation delay and likely multi-pathing confusion, especially if you are monitoring a simulcast system.
I saw firsthand a very good illustration of this just the other day; local PD was working an intersection for a funeral procession. I saw him key his mike and speak, but the words didn't come out of my receiver for a good 0.3-0.5 sec after his lips moved. Pretty odd to watch!