A week ago, I learned that it is hard(slow and awkward) to run while upright in dreams because your body is inhibited from moving. Small movements, in close proximity to your body are easier to accomplish. From that, I inferred that running on all fours in a dream is just easier for your brain to process. Running on all fours requires short movements of your appendages in close proximity to your body.

That's very interesting. It makes me think of dogs running in their sleep.

Since it seems to be normal to walk upright in dreams, I wonder if fear/anxiety (from whatever is causing you to dream about running) is an added factor making it harder for your brain to process more difficult movements.

well in my dream last night i dreampt i was running to a fight and i was up right and i could feel my chest get tight.

My personal belief is that it's just me connecting to the inner animal in me, primal if you will. I once had a women tell me i have a strong animal magnetism within me and it was an attraction to others, as i researched more i learned that in being a libra its said i am to be both alluring and hold some sort of attraction to others. Since then i've noticed ppl have alwys seem 2b willing to open up 2me n they find me very trusting ina way. To sum up i think its just an inner beast in us coming to light if u will.