Hunting on horseback was done in Germany by the nobility some centuries ago, but only for noble game: deer. It was called Parforce-Jagd. There are still the big round horns in use wich were blown throughout the hunt, called Parforce-Hörner. So it was in France, too.
Nowadays in Germany this kind of hunting is forbidden by law to prevent cruelty to animals.
In the song mentioned above Ein Jäger aus Kurpfalz the hunter orders his horse to be saddled, but nothing about chasing the deer. A line says about this game: Between his legs the deer must be shot.

Foxes are low game and to be killed by peasants. They are usualy traced down to their foxearths, then the dachshounds are sent in. By the subterranean noises the hunters learn the place where the fox defends himself and start digging. If there are too many foxes around the use of poison might be allowed by the authorities.
Following the English example of fox hunting on horseback some horse associations perform sham hunts without foxes. A scent is laid for the dogs to a certain point; first one there is the winner.

There are a lot of hunting songs in Germany (I own a fine song book published by the German Hunting Association), but I still have to meet the first song about fox hunting on horseback.