To run in Kingdom Come Deliverance (KCD) you must hold (PS4) / (Xbox One).
The game does not show you a control scheme anywhere so this can easily be missed in the early game. Running consumes stamina. Running also increases your Vitality skill over time.
You also earn the trophy “Ranger” for running 50 km in total.
If your health drops so does your stamina. To refill health you need to sleep in a bed. Then you also get your stamina back and can run for longer.