Since we've seen you take on the Hearts/Card's court is like, what do you think the chess (white and red) courts are like?
The kingdom of chess, unlike the kingdom of cards, is more fragmented; a reminder of the mirror through which it can be reached.
Similarly to the card nobles, the main chessmen—King, Queen, Bishop, Knight, and Starfleet—are human, but require magical wooden prosthetics to move around the board, which is a very arduous field.
The King and Queen have thrones, as do the bishops; as they are religious. Obviously, the Knights have knights, and finally, the Starfleet are located in what would be the rook checkmate.
Similarly to the families of Wonderland, the humans of the kingdom of the chessboard have given birth extensively (in addition to migrations).
The pawns are wood-people, who, like Cedar, are self-propelled puppets. They are built by the human families who live there, and are essentially service robots for the population.
The pawns that are not strictly human are the Checkers. The Lady of Checkers is a sort of fairy godmother/Blue Godmother, while the pawns are like the card people i.e objects that can move
The ladies are treated like the pearls in Steven Universe, that is, as servants to human families. The average human in the Chessboard Kingdom is similar to Tweedledee twins.
There are two current royal families: the Whitechecks and the Redfords. In one way or another, the two queens always manage to be related, however, they are not always the direct heirs to the throne. Sometimes they are the consorts of the White King/Red King, sometimes sisters-in-law, sometimes sisters, etc. The current Red Queen and White Queen are sisters: King Queen inherited the Redcheck throne, while White Queen married the White King and became queen.
Both kingdoms expect the heir to the throne to serve a period as a knight to prove their worth. Lily is still very young, but Chase is at the right age.
The two kingdoms are constantly at war, and the Kingdom of Hearts supports this because at least it prevents them from invading.
The current kingdoms, however, are at peace, as the two queens' brother married the Queen of Hearts. So Lizzie, Lily, and Chase are cousins.
The chessboard court is, like the card court, very similar to the Victorian one. So they are very rigid in their Wonderland madness. However, since they are two relatively small kingdoms, the average population is more affluent and upper-class than the average population of the Kingdom of Hearts. It's no coincidence that they're all so snobbish.
To everyone's surprise, the Hatter family is originally from the White Checkerboard, and Maddie and Hatta's extended relatives still live there! The White Queen is actually fond of the family.