A slight variant: After you move the piece, roll the dice 1d6: 1=Pawn, 2=Knight, 3=Bishop, 4=Rook, 5=Queen, 6=No change. The probability of the piece staying in the same kind of piece, is now 1/3. (Or, combine it with the idea of the previous comment so that 6=King)
One more slight variant: Same as above, but a piece also does not change if all of the pieces of the chess set of the kind indicated by the dice are already on the board (and pawns promote only to lost pieces).
One more slight variant: Same as above, but a piece also does not change if all of the pieces of the chess set of the kind indicated by the dice are already on the board (and pawns promote only to lost pieces).