Games might end more quickly at higher levels since one side has to sacrifice a piece for a "promoting" pawn to prevent the promotion on the 7th/2nd rank to a bishop or knight; chances are, an extra bishop or knight is sufficient for a win in higher level games.
A variant that would bring about more dynamic positions in the middle game, encouraging the sacrifice of a minor piece for a couple pawns. It would also secure a win in certain endgame positions that would otherwise be a draw in regular chess (such as the wrong coloured bishop). When a pawn promotes, it is almost always to a queen. Having the option to underpromote to a minor piece (not a queen or rook) when on the 7th rank increases the chance of underpromotion, while at the same time increasing the power of a pawn.
The only rule difference to classical chess is that pawns would have the option to underpromote to a bishop or knight when reaching the 7th (or 2nd) rank. 7th rank for white pawns, 2nd rank for black pawns. The option for a pawn to promote in this manner is lost if not done immediately (i.e. a pawn cannot sit one square away from the end of the board, then a move or more later just promote to a knight or bishop without moving).
I wonder how the dynamics of chess would change with this extra rule. Thank you!