Parallel worlds chess

Parallel worlds chess is a three-dimensional chess variant invented by R. Wayne Schmittberger in the 1980s. The gamespace comprises three 8×8 chessboards at different levels. Each side commands two full chess armies on levels 1 and 3. Level 2 begins empty and obeys its own move rules.

Game description
Each player controls two complete chess armies, initially on levels 1 and 3. White moves first. The first player to either enemy king wins the game.

Move rules
On each turn, a player may move of their, including pawns. (On the turn, however, White may move no more than  pieces.) Each piece moved can move only once in the turn. The only stipulation is that none of these moves may on the same level. (They may begin on the same level, however.)

All pieces except kings can move straight up or down one level to an empty square. (Thus captures are not permitted when moving to a new level.)

On levels 1 and 3, moves, captures, and pawn promotions follow the same rules as in standard chess. On level 2, special rules apply:
 * All pieces move as chess queens;
 * Captures are not permitted;
 * A pawn that moves to the last does not promote.

Rules for pawns that move from level 2:
 * If a pawn on level 2 moves to the owner's last rank on levels 1 or 3, it immediately promotes.
 * If a pawn on level 2 moves to the owner's rank on levels 1 or 3, on that level it can, on a future turn, move only one step to the second rank. Once on the second rank, it regains its normal two-step option.