Of course, so much of this is speculative. There seems little question that the early prog bands were aware of each other, and perhaps influenced by each other to varying degrees, either intentionally or not.
It would seem that, historically, the place to start would be the very first "prog" albums/bands, and how things developed thereafter. In this regard, I suppose we would have to include Floyd's Piper, Moodies' Days, Zappa's Freak Out, even Sgt. Pepper as influences. Possibly even Vanilla Fudge's debut album, with its proto-prog sound, particularly the enphasis on organ and keyboards. All of these (and others) pre-date Crimson, Genesis, Yes, Giant, Tull, etc., all of whom created (or released) their first albums in 1968 or 1969.
It was only after this that Genesis, Yes, Crimson et al could have influenced each other - but the foundations of their influence had to come from the earlier bands/albums, and were probably more significant than the influence they may have had on each other.
That's my story, and I'm sticking to it. LOL.
Peace.