Every round of the game takes a cut. The cut buys the share and credits it to the collection — one equal part per token, five thousand ways. Nothing is sent to anybody: each one is drawn by whoever holds it, whenever they like.
The cut collects inside the game and is handed over by an open call, then spent on the share by another. Both are open to anyone — the price they accept comes from the pools' own averages, so there is nothing to gain by calling them at a chosen moment, and nothing to wait for if you would rather not.
The contract remembers how much has ever accrued to a single token and pays each one the difference between that and what it last drew. Two things follow, both of them wanted. The collection itself needs no cooperation — it is an ordinary ERC-721 with no transfer hook, and none is required here. And a token carries its undrawn income with it when it is sold: the buyer draws it, which is another way of saying the seller was paid for it in the price.