Take care of your horse, put them to bed, age them (or wait until the next day for them to naturally age), and when they wake up they will give you a few droppings

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The best way is to start up a small farm of horses you work daily. Make sure none of them have parchments, as they won't produce droppings. Drafts are also great for this as they can eat more without the risk of becoming fat.

You can also win them in some promos.

Also, when you get Tianma (at level 30 in the trophies), make sure to work him first in the day just in case his bonus activates, so that when you wake up your horses the same day, you get double droppings from them. (It's luck-based whether you get the bonus, though, my Tianma had only activated once so far. )