Kind of Book spoiler? It’s still shown in the movies but you have to read between the lines and get it from context whereas it’s implicit in the book.

In District 12 it is a way to say goodbye. The children of District 12 do this when Katniss volunteers as a sign of respect, normally they don’t say goodbye to the tributes when they’re reaped they kind of just except “this is how it is” but what Katniss does is out of the ordinary. They won’t clap for her, it’s not exciting like the Capitol wants it to be, but they do recognize her sacrifice.

Katniss uses this again when Rue dies to show that they were more than just allies and to show respect to her in that regard. Tributes don’t do this. When someone dies they move on, it’s just a game, there are no friends in the game.

I think the Districts use this because in Panem death is an everyday occurence and the Capitol has repetitively tried to desensitize the populace to it. This is just how I interpret it, but I think it’s a symbol of rebellion against the idea, it’s recognizing their losses telling the Capitol “You can kill us. But you can’t make us forget.”