Antoine's proposal should work very well if the number of players is
prime. If the number is divisible by 2 or 3, there's a small weirdness.
Look at the Shogi pairings in the 9-player example: players 1, 4, and 7
play one another; 2, 5, and 8 play one another, and 3, 6, and 9 play one
another. No one in any of these groups plays Shogi against anyone in
another group. A similar partitioning into 2 groups would happen with
Xiangqi with an even number of players. I don't know how much, if at
all, this should bother us.
Here's an interesting possibility, inspired by Fergus's idea of having a
champion for each game. Perhaps the final round could consist of the top
Chess player, the top Xiangqi player, the top Shogi player, and the top
overall player (and, if some of these should be the same person, the 2nd
overall player, etc.).
I would prefer to allow draws by agreement even in Shogi, although they
should be discouraged except in clearly drawish positions. It doesn't
seem fair to me to penalize both players for playing equally well just
because the game ran long. If length is a concern, there must be some set
of carefully chosen time controls that will address the issue.