There is no zakah on horses and donkeys if they are not for trade

The animals on which zakah is due are the an’aam animals, i.e.: 

1-Camels

2-Cattle

3-Sheep, which includes goats

That applies if they reach the nisab and are pastured, i.e., if they graze and eat of the plants that grow on the land, and their owner does not spend anything on buying feed for them. See the answer to question no. 93387 and 40156. 

As for other animals, no zakah is due on them unless they are prepared for sale. 

Al-Nawawi said in al-Majmoo’ (5/311): The Muslims are unanimously agreed that zakah is due on camels, cattle and sheep. As for horses, mules, donkeys, etc., no zakah is due on them. This applies if they are not for trade; if they are for trade then they are subject to zakah, because of the report narrated by Abu Hurayrah (may Allah be pleased with him), that the Prophet (S) said: “The Muslim is not obliged to pay zakah on his slave or his horse.” End quote. 

The scholars of the Standing Committee for Issuing Fatwas were asked: Some people spend the wealth that Allah has spent them on keeping valuable purebred horses the price of which may be thousands of dinars in order to enter races with the aim of winning the prizes offered. My question is: Are these horses and their offspring subject to zakah, and what is the nisab and the rate of zakah that is due?  

They replied: 

If the matter is as described, and they buy them to keep them, not to sell them, then no zakah is due on them, because the Prophet (peace and blessings of Allah be upon him) said: “A man does not have to pay zakah on his horse or his slave.” Saheeh – agreed upon."(Fatawa al-Lajnah al-Daa’imah  9/216). 

And Allah knows best.