I worked with horses for a few years when I was young, mostly racehorse farms (that was the only
local employment where I lived). Bringing the breeding stock in and out from the paddocks was part of my job and I would do anything to avoid the broodmares and foals. I got bitten so badly, so often, that I took it as a given I was going to get bitten. Generally in the small of my back or in the butt. To this day I see a foal and my first thought is to watch its muzzle rather than "oh how sweet".
Like Joshua says, orphaned foals were even worse- usually picked up all sorts of bad habits since they had no other peers to bite them back and teach them that roughhousing has its limits.
I don`t know what the plans are for the donkey but when you`re working with horses you want to avoid anything that will make them sensitive about their head in the future, so flapping your arms or yelling or swatting them is probably not a great response.
With a foal, (very much like the video describes, it`s funny) I was taught that after a bite you pinch/twist the bottom lip (IMMEDIATELY, or they won`t get it) like another foal would do to discourage them. And like the video, i preferred to turn it into scritches/petting instead. The thing is, the foal needs to be halter trained, because if it`s bouncing all over the place, you`re not going to be able to keep track of where its head is and you`re going to get bit. Maybe halter training the foals also distracted them from wanting to bite, who knows. Again, though....this is what I would do for a horse, I also don`t speak donkey.