Single Comment

The rules are a little sparse. While no rule specifies, I assume a king
can't move itself off a board, although that could be interesting. Maybe
if they were on the same square on adjacent boards, or adjacent squares on
the same board, a king could move another king to a different level. [Let
the players move a piece per king, only the board the king is on.]  
Understanding how the unlimited sliders move 'through the kings' is
easy, with your diagram [once you figure everything out], but what about
pawns and knights? Does the pawn have to be able to move 2 squares to go
through to another board? If it doesn't get 'stuck inside' the
receiving king when it can move only 1 square, and it can move through
both and come out in front of the 2nd king, then it can capture by moving
diagonally through, and what about the knight? If a pawn can move through
a king by landing on it, why can't a knight? So, where does it come out,
in that case? If it must be next to the king, does it have to start only
orthogonally adjacent and end diagonally adjacent to the 2nd king, or can
it also start diagonally adjacent to the 1st king and end up orthogonally
adjacent to the second king?
Interesting big variant, needs some work to be more than an idea.