The question is not really clear. "core muscle" weakness is not really a thing related to running nor is it important to running. To answer what are the bio-mechanical consequences will be a logical leap as i am not aware of any prospective studies that have examined this issue.
Any deep core muscle weakness should be taken into heavy consideration when planning exercise. Imagine how much strain is put on the spine when you begin to anterior tilt the pelvis or nutate the sacrum. Running only magnifies this load and can become damaging very quickly