Pigs should be considered farm animals as they are relatively docile, and will not run away from the player should they approach.
Stand close while not holding anything and a screen prompt 'Carry Animal' will show. Press 'F' to pick up the pig. You can not carry a pig in a vehicle. Pigs despawn in small enclosures. Entering your inventory, taking out a weapon or eating and drinking will drop the pig. Opening a door causes the pig to get attached to your backpack.
Pigs can drop: