I volunteer for a rescue who finds people stuck up mountains and are lost in very rural places moors etc. Many have a medical emergency. Frequently they have dogs with them, many of which will be found some way from the person as the anxiety the dogs feel makes them want to reverse their movements and go home. Others will run at the sound of us approaching, which often will be a single rescuer as we split up in some areas, to cover more ground.
Neither myself or the team have ever had to chase after a dog. They’re without fail, overjoyed to see us. They’re pack leader isn’t able to lead, so they are very, very relieved to have another one.
Dogs are dogs.
Just my several decades of experience including some high profile cases.