Thanks for your advices Lydia and Amiga.
I will try both garlic and apple cider vinegar (not sure if I can find it in my place). Today, he's not lying anymore. He's walking and running as often as he was before. But he still has a small limp. I checked and he doesn't seem to suffer from a bumble foot (not 100% sure yet) but I rather bet he's recovering from a strain or a pull
Amiga, about runner (and kaki) body language, after mating, my ducks run. They just run very fast, in a crazy way and lowering the neck. I find it hilarious but is it the same for all ducks?