Did the GD invent the concept of song transitions?

Did Jazz players ever do this? Trying to think of an example. 

I know they would throw nods to other songs in, runs of notes from other composition, etc - but segues? I'm not sure.

Great question - and hopefully an enlightening thread as it unfolds.

They're not live performances, but Sgt. Pepper's and the White Album have quite a few segues from one song into another.

I saw the Beach Boys ca. 1963 - my first rock show, age 12, had to have ny dad drive me - and they did several "medleys", as they were then called, of some of their hits. Each song was usually shortened, but they did run right into one another. Other bands and shows from this era also included medleys.

"Devil with a Blue Dress On / Good Golly Miss Molly" by Mitch Ryder and the Detroit Wheels (1966) has a place here too.

One could make a strong case the symphony is a bunch of songs known as movements that transition into the other movements. 

That's a two song medley, Mike.  Lots of those, before and after the Dead:

https://en.m.wikipedia.org/wiki/List_of_musical_medleys

The Mothers Ot Invention did this 3 song segue on their 1966 debut, "Freak Out":

Help, I'm a Rock (Suite in Three Movements)

I. Okay to Tap Dance

II. In Memoriam, Edgard Varèse

III. It Can't Happen Here"

I think this was being done by Fife and Drum bands in the Revolutionary and Civil Wars, and am fairly certain it was commonplace in Appalachian Folk Music and it's antecedents in British Folk and Celtic Music.  I've heard medleys in Gospel Music and New Orleans Marching Bands too.

Brazilian music would use segues to keep the beat going and people dancing.  From Wiki:

"In some Brazilian musical styles, where it is called "emendar" ("to splice"), in particular in Samba and Forró Pé de Serra, it is very commonly used in live performances, creating sets that usually last around 20 minutes but can sometimes take more than an hour, switching seamlessly between different songs. The larger rhythm groups of bands, with up to ten percussionists in Samba for example, facilitate the switching of one song to another, as the percussionists keep the rhythm or beat going while the pitch instruments prepare the harmonical transition to the next song, often with just one pitch instrument leading this transition. In Forró trios, where the only pitch instrument (apart from the voice) is the accordion (which plays together with two percussionists), the accordionist usually "puxa" ("pulls") the next song as soon as the previous has finished."

https://m.youtube.com/watch?v=0DHOJikaj44

Red Nichols "New Orleans Medley" (1932)

what do we mean by 'transition'?

are we supposing 'play an arranged transition from one song to anther' or 'jam out into something out side the original song structure/formlessness (spelled out, that's a weird fuckin' word) then find your way back into a song'?

these are different things.

As far as "seamlessness", if you will, it's hard to think of anybody even close.