If we beat other competitors, it promises material benefits and personal success. If our competitor in our race is an animal, it means victory, but we have to see the meaning of the animal to know about what we win. If we see a person is running, then it indicates unknown events. If it is about several people, then it represents disputes and quarrels. If they are armed, it announces insurrections, riots or wars.
Read more about dreaming of Running in other dream meanings interpretations.