I started getting into chess lately, and went back to my copy of Through the Looking Glass to see how the chess game played. Alice of course starts as a pawn and becomes Queen Alice on the other side of the board. It turns out to not be a very good game, the Red King does not move out of check, which is illegal to do! The characters though of course act erratically in the book, so it is a good game in terms of that characterization. The Queens fly across fields, relating to the powerful queen pieces in chess, while the kings are pretty anxious and immobile. The knights falling off their horses mimics the L-shape movement of knight pieces. When Alice is on the same row as another character, she is talking with them. It is good to be reminded how clever the original books are!
I found this animation showing the chess game, which I think is helpful:
It's from this site which has good information about the game in relation to the book.
If you have further interest in this, you may want to look at the notes in Martin Gardner's Annotated Alice