That clean first shot that strays away from the following shots also has another difference - the barrel is completely cold at that point. My White .45 doesn't seem to care at all whether the first shot is from a fouled bore, but that first shot will strike just a tad lower than the rest due to the cold barrel. Keep up a rhythm and it just lays 'em in there but let it cool completely and it will hit a bit low again even though it's still fouled. The difference isn't enough to cause a miss on a large game animal though, at least not at reasonable range.

Just an observation, may be relevant to some and not to others.