I'm not entirely sure what exactly it is you're referring to, though I think I have a good idea, but if you could find an example that may help others answer your question. That said, without knowing exactly what you're speaking of, I think it might be reasonable to point out that those "echos" might actually be coming from speakers placed further away from the person giving the speech. Often, when delivering speeches to large crowds, you would have many speakers (the electronic kind, like from a stereo, not as in other people speaking) positioned throughout the crowd and it would relay what was being said so that those further back could hear more clearly. While the speakers may be playing the same sounds almost at exactly the same time, the distance between them can result in an echo.

More sophisticated setups can even take into account the difference in time and speed of transmission of electronic signals vs the speed of sound, compensate for them so as to reduce the echo effect, etc, but that's straying a bit off topic.

What I'd be interested in knowing is if anyone has purposely exploited this effect for dramatic effect on purpose. After all, it's quite common to pause for dramatic effect when giving a speech, so it'd be interesting to know if speechwriters and public speakers (the human kind, not the stereo kind) have ever purposefully chosen to exploit this by say, specifically choosing certain words over others to reverberate and echo, or things like that, or if it's just a happy coincidence in many cases leading to the dramatic effect of it.