"to go in procession," 1814, "A colloquial or humorous back-formation" from procession [OED]. Accent on second syllable. The earlier verb was procession (1540s).
early 14c., proces, "fact of being carried on" (as in in process), from Old French proces "a journey; continuation, development; legal trial" (13c.) and directly from Latin processus "a going forward, advance, progress," from past-participle stem of procedere "go forward" (see proceed).
Meaning "course or method of action, continuous action or series of actions or events" is from mid-14c.; sense of "continuous and regular series of actions meant to accomplish some result" (the main modern sense) is from 1620s. Meaning "a projection from the main body of something," especially a natural appendage, is from 1570s. Legal sense of "course of action of a suit at law, the whole of the proceedings in any action at law" is attested from early 14c.; hence due process "fair treatment" at law, considered as a right (mid-15c.).
1530s, "begin legal action against, summon in a court of law," from French processer "to prosecute," from proces (see process (n.)). Meaning "prepare or treat by special process, subject to special process" is from 1881, from the noun in English. Of persons, "to register and examine," by 1935, in reference to the U.S. Army. Related: Processed; processing.