Some people believe that some microphones are better (or worse) at picking up sound as the distance increases. In general terms this is not the case for conventional live sound microphones. Remember ... microphones do not go out and search for sound, the sound comes to them. To do otherwise would require that some kind of amplification that could kick in for a distant sound and that the mic could tell the difference between distance and SPL. It is the sound itself that falls under the inverse square law. IOW it's your voice that is dropping off in feeding the mic.
The example given was pulling a mic away a foot and having it go dead. So here's how that breaks down.
Just to keep it simple we'll stay in tens. If you start singing at .1 foot away and that produces 100 dB spl (wherever you are measuring it) then moving away from your mic to 1 foot the signal would drop by 20 dB to a new spl of 80 dB. This is the normal loss due to inverse square from the source over distance. Microphones are simple but dumb devices and cannot distinguish distance from simple lower level. If this were the case then the mic would become unusable as you sang softer. They just convert, in a reasonably linear fashion (at least at these distances) the pressure on their elements to voltage. It makes no difference whether the element is dynamic or condensor.
Now if you factor in proximity effect you will see a bit of an exception to this ... but only to the frequencies below about 200 Hz. I have made the measurements at 18" compared to 1" and can tell you at that distance an SM58 will exhibit a bump of about +10 dB at aprox 120 Hz. An OM-5 will exhibit less bump ... about +7 dB at about 180 Hz. (the bump being at the closer distance of course)
So back to the claim that the Audix picks up less at distance ... Compared to an SM 58, they both pick up the same at distance for the normal voice range of 300-3kHz but the Audix picks up about 3 dB less than the 58 at 18" but only below 200 Hz.. But that is 3 dB compared to 20 dB and not very significant. Likely another dB at 1 foot instaed of 18". It's not the exact level of the bump that is important, just the delta between the two.
So what about "shotguns" picking up at great distances? They don't pick up on axis any better than anything else. What they can do is pick up less off axis spill so the intended to un-intended ratio is better, but the signal itself is no stronger. If you are trying to pick up birds on the wire across the street with a shotgun compared to a 58 you'll get the same level of birds in both mics, but you'll get less traffic noise mixed in with it with a shotgun.