Speaking of how other things influenced jazz and how jazz influenced other things, here's an interesting brief essay entitled "Learning to Swing" from Ralph Ellison's book Shadow and Act. That essay is included in Daryl Cumber Dance's anthology From My People-400 Years of African American folklore (New York; W.W. Norton & Company; 2002; p 195) :

"You see jazz was so much a part of our total way of life that it got not only into our attempts at playing classical music but into forms of activities usually not associated with it: into marching and into football games, where it has since become a familiar fixture. A lot has been written about the role of jazz in a certain type of Negro funeral marching, but in Oklahoma City it got into military drill. There were many Negro veterans of the Spanish-American War who delighted in teaching the younger boys complicated drill patterns, and on hot summer evenings we spent hours on the Bryant School grounds (now covered with oil wells) learning to execute the commands barked at us by our enthusiastic drillmasters, And as we mastered the patterns, the jazz feeling would come into it and no one was satisfied until we were swinging. These men who taught us had raised a military discipline to the level of a low art form, almost a dance, and its spirit was jazz."

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Here are two definitions of "swing" from http://en.wikipedia.org/wiki/Swing_jazz

"Swing music, also known as swing jazz or simply swing, is a form of jazz music that developed in the early 1930s and had solidified as a distinctive style by 1935 in the United States. Swing uses a strong anchoring rhythm section which supports a lead section that can include brass instruments, including trumpets and trombones, woodwinds including saxophones and clarinets or stringed instruments including violin and guitar; medium to fast tempos; and a "lilting" swing time rhythm. Swing bands usually featured soloists who would improvise a new melody over the arrangement. The danceable swing style of bandleaders such as Benny Goodman and Count Basie was the dominant form of American popular music from 1935 to 1945.

The verb "to swing" is also used as a term of praise for playing that has a strong rhythmic "groove" or drive." "

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So when Ellison wrote that no one was satisfied until they were "swinging", he meant that second definition-"playing [making music] that has a strong rhythmic "groove" or drive."