I heard it a lot more in English than in Italian but it seems it's generally agreed that it is an old Italian proverb. It appears described as such in several books written in English, from various parts of the world (one example I found was an Indian journal). One of the oldest, "The Westminster Papers: A Monthly Journal of Chess, Whist, Games of Skill and Drama", Vol. 6, dates 1873-74 and writes "We shall all follow, cousin, that is certain, whether we be first rates, or lowly recipients of the Queen 'At the end of the game,' says the old Tuscan Chess proverb, 'the King and the Pawn both go into the bag.'"
Normally only king and pawn are mentioned as "Alla fine del gioco, re e pedone finiscono nella stessa scatola" (Once the game is over, the King and the pawn end in the same box)... once dead, rich and poor are in the same situation.
You might find tornano rather than finiscono and other small variation in wording but the meaning is always the same and the chess pieces are just the most and least valuable in the group, adding the rest just makes a mess of the meaning.
Easily enough, some variation in wording could be due to translation from the different dialects used in Italy. It seems to be a Tuscan proverb however.
As for the real origins it could be that the proverb wasn't even born in Italy, it might be that it's an Indian/Persian/Arabic proverb that simply followed the game in its history.
During an exchange in the comments, looking for references I came across a book called "Miscellanea sul giuoco degli scacchi". It appears to be a collection of texts about the game of chess by various authors. At the bottom of page 61 they show a sonnet by Francesco Bracciolini (1566 – 1645) dedicated to Lena Fornaia that reads "Un proverbio toscano dice: Alla fin del gioco tanto va nel sacco il Re quanto la pedina".
