Jogging with your dog is an incredibly rewarding and beneficial activity for both you and your dog. It can help establish a strong bond, reduce stressful or fearful behavior issues, and improve overall health. In my most extreme case, rehabilitating Tigger, it meant I could walk her around the neighborhood.
Before jogging, Tigger would fly to the end of her leash in a sheer panic upon hearing anything, anybody, or seeing any moving item (car, bag blowing in the wind, a leaf falling to the ground, a person two blocks ahead of us). Jogging meant that she was distracted enough to deal with everything, within two outings. She would be mildly aware of the things that made her panic, but she would just look at them and keep moving. Eventually, she would be relaxed and wouldn’t be the least concerned about them. Eventually (about four months later) this transferred to walking, too.
I believe that fearful dogs, especially ones that panic, can benefit the most from picking up the pace and jogging. It distracts them enough that they can learn to pass by the triggers without panicking. Repeat this enough times, and they can adjust to the world without the need to panic.
Advantages for the dogs:
Jogging particularly helps fearful dogs and dogs who are poor leash walkers. Moving at a faster pace helps distract a shy dog from perceived threats (other people, loud noises, other dogs, and so on). The dog concentrates on the direction, speed, and movement of the jogger more than the environment around them. This same reason means that jogging can also help poor leash walkers (pullers, stubborn walkers, rampant “sniffers”), as they should move more consistently and pleasantly on the outing.
The additional exercise can also help dogs become more balanced behaviorally, and reduce overall stress and boredom levels. Dog guardians have reported having fewer issues with barking, “OCD” like behaviors, digging, fearful behavior, and more.
Lastly, dogs have a natural affinity for an increased pace (such as jogging or running). For many dogs, it is a natural and easy gait that they enjoy to move at, and they will greatly enjoy activities where they can exercise at this speed. Not to mention, it can help file down their toe nails a bit (or a lot, with soft nailed dogs!)