If you have a non-runner before the start of the first race, you can edit your stable as many times as you like, changing the non-runner for another horse or changing a number of horses to allow for a different makeup of your stable.
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If one or more of your horses are declared non-runners after the game has started, the non-runner will be automatically replaced in your stable by another runner in the race. You will be given a horse of equal value if available and if more than one is available, you’ll get the one with the lowest racecard number. If there isn’t a horse of equal value, you will get a horse of nearest guinea value, again ordered by racecard number if there are multiple horses at that price. If there are no horses of equal or less value, the nearest horse of higher value will be subbed in.

Non-runners will be indicated in your stables like this: