Carnival of the Animals

I have added a subvariant of this, and a few other variants with dice in a similar vein.

Setup

Pieces

Firstly the Pawns are Eurofighter Pawns. The initial double-step move can be either two noncapturing steps both orthogonally forward, two capturing steps both diagonally forward (not necessary the same diagonal), or one of each in either order. If the intermediate cell is en prise to an enemy Pawn, which is not itself captured on the second step, the enemy Pawn can immeditately capture it En Passant. A piece captured on the second step is not recovered by an En Passant capture.

Secondly some pieces mutate through the rolling of dice.

Rules

Each player's first move possibilities are the same as in FIDE Chess, except that it ends by that player rolling one or two dice to determine what their next move can be. Subsequent moves also finish with rolling dice. Note that they are always rolled half a move in advance, so that the opposing player can avoid check by the pieces as they are when they next move. If a roll gives the player a choice or mutations, the choice must be made before the opponent moves for the same reason. The alternative of effectively making the pieces compound would make them too strong. Keeping track of each player's roll during their opponent's move is why not sharing dice is better.

Pawns reaching the far rank are promoted after the dice roll to Rook, Knight, Bishop, or Queen. How they move in promoted form is determined by the dice roll. No other piece can be promoted, however it moves to the far rank.

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CARNIVAL OF THE ANIMALS

Only Knights mutate. A player rolling two numbers less than 6 has their Knights turned into the piece whose leap is represented by those two numbers. It has a 1 in 18 chance of being each (total over all oblique pieces 5 in 9) of the 10 oblique pieces:
2:1 Knight;
3:1 Camel;
3:2 Zebra;
4:1 Giraffe;
4:2 Charolais;
4:3 Antelope;
5:1 Zemel;
5:2 Satyr;
5:3 Gimel;
5:4 Rector.
It has a 1 in 36 chance of being each (total over all diagonal pieces 5 in 36) of the 5 diagonal pieces:
1:1 Ferz;
2:2 Elephant;
3:3 Tripper;
4:4 Commuter;
5:5 Quitter.
Links are to the piece itself where it has its own Piececlopedia entry, otherwise to the article in which it is named.

A player rolling a 6 and another number can choose which number between 1 and 5 the 6 represents. They have a 1 in 18 chance of getting each of the 5 combinations of 6 and another number (total over all such combinations 5 in 18), which all offer the choice between 4 oblique pieces and 1 (italicised below) diagonal one:
6 and 1 a choice between Ferz/Knight/Camel/Giraffe/Zemel;
6 and 2 a choice between Knight/Elephant/Zebra/Charolais/Satyr;
6 and 3 a choice between Camel/Zebra/Tripper/Antelope/Gimel;
6 and 4 a choice between Giraffe/Carriage/Antelope/Commuter/Rector;
6 and 5 a choice between Zemel/Satyr/Gimel/Rector/Quitter.

A player rolling two sixes can choose any of the 15 pieces. Again this choice must be made before the opponent moves. There is of course a 1 in 36 chance of this happening.

As Kings and Rooks are unaffected, so is castling. One thing to note is that the King must not move into, out of, or through check by whatever leaper the opponent's dice roll has just produced as well as check by Queen, Rook, Bishop, or Pawn.

If a player is sure that they will not be moving a leaper next move, or promoting a Pawn to one at the end of the current move, they need not bother rolling.

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CARNIVAL OF THE ANIMALS, ALL OBLIQUE SUBVARIANT

In this subvariant 6 is treated as a normal number, and it is rolling a double number that gives the player a choice. The extra possibilities from rolling a 6 and another number are the:
6:1 Flamingo;
6:2 Crane;
6:3 Chamois;
6:4 Zherolais;
6:5 Parson.

A player rolling a double number can choose which other number the duplicate represents. They have a 1 in 36 chance of getting each double number (total over all such combinations 5 in 36), which all offer the choice between 5 pieces:
double 1 a choice between Knight/Camel/Giraffe/Zemel/Flamingo;
double 2 a choice between Knight/Zebra/Charolais/Satyr/Crane;
double 3 a choice between Camel/Zebra/Antelope/Gimel/Chamois;
double 4 a choice between Giraffe/Charolais/Antelope/Rector/Zherolais;
double 5 a choice between Zemel/Satyr/Gimel/Rector/Parson;
double 6 a choice between Flamingo/Crane/Chamois/Zherolais/Parson.

If a player is sure that they will not be moving a leaper next move, or promoting a Pawn to one at the end of the current move, they need not bother rolling.

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UP AND DOWN CHESS

This uses just one die aside, and mutates all pieces except Pawns. Rolls are interpreted as follows:
1 no backward or sideways orthogonal moves, weakening Rooks (including to prevent Castling) and, to a lesser extent, King and Queen;
2 no backward diagonal moves, weakening Bishops and, to a lesser extent, King and Queen;
3 no Knight moves except the two forwardmost, reducing Knights to Helms;
4 Knights gain Calf move (2 forwardmost 4:2 leaps) to become Nightpigs, a piece also used in my Year of the Pig Variants;
5 Bishops gain Wing (FO Rook) move to become Silveriders;
6 Rooks gain Mitre (FO Bishop) move to become Goldriders.

These effects last only for one move and the next roll, if different, reverses them.

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PIECE ORIENTATION CHESS

This uses two distinguishable dice. The first die determines the piece type affected:
1 Pawns, always preventing double moves and En Passant;
2 Rooks, always preventing Castling;
3 Knights;
4 Bishops;
5 Queen;
6 King, always preventing Castling.
The second determines how they move:
1 as piece with corresponding Shogi starting place, ignoring the existence of actual Bishops and Rooks in Shogi;
2 as piece with corresponding Xiang Qi starting place, though without River or Fortresses;
3 as compound of 1 and 2;
4 as piece with corresponding Makruk starting place;
5 as compound of 1 and 4;
6 as compound of 2 and 4.

Notable consequences of combinations of rolls are as follows, noting that order is important:
1 and 5/6 gives Pointpawns, able to capture on any forward radial;
2 and anything but 1 has no effect;
3 and 3 gives a Helmed Mao, blockable except in the two forwardmost directions;
4 and 3/6 gives a Silverfearful, that can move one or two steps diagonally or one forward;
5 and 3/6 gives a Prince;
6 and 2 reduces the King to a General.
2/3 and 1 cannot be applied to pieces on the enemy back rank.

Again these last for just one move. Pieces other than that determined by the first die have their FIDE moves, including Castling, except that Pawns have the extra Eurofighter moves. Shogi movement neither removes the full range of FIDE promotability from Pawns, nor confers Shogi promotability on other pieces.

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ARMY ORIENTATION CHESS

Uses a single die, similar to the second in Piece Orientation Chess but applying to the whole army and not drawing on Makruk. The rolls are:
1 as pieces with corresponding Shogi starting place - the existence of actual Bishops and Rooks in Shogi is ignored;
2 as pieces with corresponding Xiang Qi starting place, though without River or Fortresses;
3 as compound of 1 and 2;
4 as FIDE, but with Eurofighter Pawns;
5 as compound of 1 and 4, differing from 4 in having Silverbishops and Pointpawns;
6 as enhanced in 4-6 of Up and Down Chess.

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MULTIPAWN CHESS

Instead of always having Eurofighter Pawns, the type of double-step initial move that Pawns can make - and which enemy moves they can respond to with En Passant - is determined by rolling a die. A full list of double-step rules for Pawns and similar pieces may be found in my piece article Man and Beast 02: Shield Bearers.
1 gives ANCIENT PAWNS, no double-step move at all;
2 gives EUROPEAN PAWNS, both stages much be non-capturing;
3 gives WARHEAD PAWNS, both stages must be capturing;
4 gives TRIDENT PAWNS, both capturing or both non-capturing but not one of each;
5 gives HELMSMAN PAWNS, one stage capturing and one non-capturing in either order;
6 gives EUROFIGHTER PAWNS, unrestricted.

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MONTY PYTHON DRAG CHESS.

In this, the six piece types represent the six main members of the Monty Python team. King: John Cleese (tallest), Queen: Graham Chapman (Cleese's writing partner), Rook: Eric Idle (relative loner, and Rooks are furthest apart), Bishop: Terry Gilliam (cartoonist can better represent diversity of names in different languages than live actor), Knight: Terry Jones (something of a "workhorse" of the sketches), Pawn: Michael Palin (from his personal humility). For the first move each player moves a Knight or Pawn normally. Each player follows their move (first or subsequent) with a dice roll determining which of their pieces is in drag next time. It is done before the opponent's move to avoid check and anticipate capture. In subsequent moves the player either uses the Drag Move of the piece whose number was rolled, or moves another piece normally. Drag Moves are as follows, with their dice roll numbers:
1: Michael Palin was always especially self-effacing about his drag capabilities, so the Pawn's Drag Move is as a Silvergeneral, among the weakest retractable supersets of the Pawn's normal move. Eurofighter moves remain possible, even to the point of one Drag-Move Pawn capturing another En Passant!
2: Eric Idle was the most convincing of the six in drag, so the Rook's Drag Move adds the Ferz move - a "drag-on-king" to use a pun on the enhanced piece's Japanese name, and little weaker than the strongest piece out of drag. Castling remains possible if it would have been anyway, as the Drag Move is a superset of the normal move.
3: Terry Jones in drag was the "mother of all battleaxes", so the Knight's Drag Move is a reasonably powerful one, combining those of the Bishop and Wazir. This continues the pun by turning the "horse" into a "drag-on-horse"!
4: Terry Gilliam's talent was in drawing, rather than playing, caricatures of women, so I decided that the Bishop's Drag Move is to "draw" (or indeed "drag") an enemy to an adjacent empty cell threatened by that enemy and then capture it, or to move one step without capturing if no enemy piece is threatening that cell. For example, moving from c1 to c2 could capture a Rook on c5 or d2, a Bishop on b1 or a4, a Queen on any of those, or a Knight on a3 or d4 (though not more than one at once). An enemy King adjacent to a possible destination is in check.
5: Graham Chapman's drag character was basically a pale imitation of Terry Jones' one, so the Queen's drag move is actually far weaker than its normal one, that of the Goldgeneral.
6: John Cleese was always unconvincing in drag, but his height made his colleagues' drag a little more convincing when he was normally dressed, so the King's Drag Move is that it cannot itself move except to Castle, but any adjacent piece of the same army has a choice between its own Drag move or its normal move.