Pieces may move from one board to the other, but only by moving onto squares which contain chutes or ladders.
See the Composed Example which follows the rules; it will clarify this situation.
The second part of the move is optional; you may choose to neither add nor remove any chutes or ladders.
In order to be lifted, a piece must stop on the ladder square; moving across the square (as, for example, a Rook might do) does not lift the piece.
A King may not move onto a square which is attacked, even though a chute or a ladder would then take it to safety. For example, White King 0d6, Black King 0d8, ladder d7; Black may not play Kd8-d7 even though the King would be lifted to the safe square 1d7.
The player who made the chute or ladder owns it in this sense, and there is a limit on the number of chutes or ladders a player may own.
Ownership does not affect lifting or dropping. Pieces of either color are lifted or dropped equally by chutes or ladders owned by either player.
You cannot place a chute or a ladder on a square which already has a chute or a ladder.
Suppose we have Black Kings at 0e4 and at 1f5, Black Pawns at 0f4 and 1f4, a chute at e3 owned by either player, a White Rook at 0e1 and a White Pawn at 0e2. The remainder of the position is unspecified.
Now White plays 1. 0e2-0e4/+0e4, in other words the White Pawn moves from e2 of board 0 to e4 of board 0; then as the second part of White's move a ladder is added which lifts the Pawn from 0e4 to 1e4.
Now both Black Kings are in check. Fortunately, there is a saving move.
Not legal is 1... 1f4x0e3 en passant. That is, the Black Pawn on f4 of board 1 could try to capture the White Pawn en passant and then use the chute on e3 to descend onto board 0 and block the check from the Rook; but the White Pawn did not cross 1e3 and therefore the Pawn on board 1 cannot capture it en passant.
Instead, 1... 0f4x0e3 en passant removes the Pawn from 1e4 and blocks the check of the Rook at 0e1, and saves both Kings.
This example is the shortest possible game ending in checkmate.
I believe that it is, although it seems to be much more difficult than K+R v K in FIDE Chess.
One technique is to leave one enemy King free and use both of your Kings to herd the other enemy King towards a corner; the defender can squirm with chutes and ladders, but eventually the combined threats of mate and double attack should win.
Two Kings plus two Bishops versus two Kings may be impossible to win, but two Kings plus two Knights versus two Kings should at least be worth playing.
This simple endgame is remarkably complex.
1. 0d6-0d7+/+0e8 creates a ladder on the Pawn's promotion square. The idea is that Black has no choice but to move K0e8-1d8, leaving no Kings on the same board as the Pawn, and then White can play 2. K0e6-0e7/-e8, defending the promotion square and promoting the Pawn.
However, Black plays 1...K0e8-1d8/+0e7, and the newly-created ladder at e7 keeps White from playing Ke7!
In response, White might try 2. K0e6-0d6/, but K1e8-1f8/+0c7 defends all threats.
Therefore, 2. K0e6-0d6/+1c7, K1e8-1f8/+0d7; by owning the ladder on e7, Black keeps control of which board the Pawn is on.
3. K0d6-0c7/+1c7, K1f8-1g7/+1e6 4. K1e6-1d6/+1d6; White needed to bring a King back to board 1 in order to defend the Pawn.
4. ... K1g7-1f6/-e7 (removing an unnecessary ladder, Black gets below the limit and can place another ladder at will).
Black now draws easily. The Kd8 stays put, the Kf6 wanders, and Black will never remove the ladder from d7.
Perhaps a smaller board or weaker pieces would help.