I'm trying to figure out a way to move cubes around a hexagon grid (37 spaces) so that they get relatively evenly distributed among the spaces during the game.

More specifically, I want the game to start with a cube on each of the 37 spaces. Players will be able to clear out these spaces during the game but I don't want the cubes to be removed. Instead, they will move to adjacent spots with other cubes possibly on them. Each space will have an arrow(s) pointing to one of the adjacent spots that they will move to. Can you think of a way that the cubes move fluidly around the board throughout the game without them clumping up too much?

That does look close to what I am looking for. Thanks for the reference.