Political slang refers to handshaking during a campaign. One could "press flesh" as candidate or on behalf of candidate or on behalf of political cause intending to garner support from those whose flesh is being pressed.
Shaking hands while attempting
To exert force or pressure.
To weigh heavily, as on the mind of another.
To advance eagerly; push forward.
To require haste; be urgent.
To assemble closely and in large numbers; crowd.
To employ urgent persuasion or entreaty.
To encourage (a falcon, for example) to participate in the chase by feeding it flesh from a kill.
Middle English pressen, from Old French presser, from Latin pressre, frequentative of premere, to press. -Press
Middle English, from Old English -flesh
1 Volunteers press flesh and pass pens to get Nader on the ballot.
2 State workers press flesh for better pay.
3 Tom CruiseC and Japanese Prime Minister Koizumi press flesh while in town to promote the forthcoming Last Samurai.