Donkey - Flat Racing & Harness Racing
Mostly common as a festival activity and probably just general shenanigans. Locations obviously vary but it seems that it's most commonly run in the UK, Jamaica, Croatia and Italy.
Donkey racing in Italy
Donkey derby hosting 101 for the UK
Apparently it's so big in Jamaica the parliament approved 500k worth of funding for donkey racing
Less popular but a big tradition in Karachi, Pakistan. Donkey cart races are organised by the Karachi South Donkey-Cart Association (KSDCA) which has over 800 owners registered with them.
Minor races take place every alternate weekends in the afternoon with the grand events taking place in the evenings and covering a distance of seven kilometers. The grand race consists of 30-40 racers with the prize-money ranging from Rs 25,000 to Rs 40,000(about $350-550).
maybe not the safest thing to do, but here's a video
I have no issue with adding this, but same as with pony racing I will leave this open a while longer to hear any dissenting opinions!