I am a runner, I had slight lower back pain before running that went away in the early days of running. I mostly assumed it was losing weight, which I am sure helped, but there is no doubt both my focus on posture and actual posture improved. So it is a bit of a positive feedback cycle.

To your point about no to much running, after I began running for periods over 2 hours, I would get some back pain during the run and stiffness after. However, I began doing pushups and planks, sure enough I strengthened my core and I don't get backpain on even 4+ hour runs (not to mention I developed a 6/8 pack which I never had in my life, a nice surprise in the mid-30's).