Let's start off with an important definition: the "core" muscles are more than just your abs. I consider it (the core) to include anything that is used in the process of stabilisation during the act of running. So "core" therefore includes your abdominals, your back, your butt - basically anything below the neck and above your mid-thigh that isn't an appendage (so arms and legs are out, as are man parts if you have them.)
The role of the core is to provide stability to your running gait - the more stable you are, the better able your body is to turn the potential energy stored within your driving muscles into the kinetic action of running. Instability creates excess movement which is energy-potential wasted. Physics dictates that the output force can only be as strong as the resisting force (Newton's 3rd law for you science majors) - if you aren't providing a stable reaction you can't possibly generate your maximal output force. Won't happen.
There's also the word "mobility", which is a linked concept for running. Mobility is more than just being flexible - it's about being stable through your range of motion. Combine core stability with a good measure of mobility through your hips and lower back and suddenly you create an opportunity for your body to come close to meeting its potential.
So how does one prepare the core to run long distances? In my opinion there are three components, each of which is no more or less important than the others:
BASE STRENGTH This isn't necessarily movement specific to the action of running. You are simply providing the muscles with a measure of strength which you then apply the running specific action to. I prefer dynamic movements for this stuff rather than static (think bear crawls as opposed to planks) and/or large muscle group movements (squats, deadlifts etc). I also believe bodyweight exercises like glute bridges, fire hydrants and donkey kicks are important for this base strength.
It's my opinion that this underlying base strength becomes more evident the longer you run. If you can develop endurance in your postural muscles, you enable your body to hold form (and maintain output) for longer.
MOVEMENT SPECIFICITY This is where you refine your strength to apply it to building running specific support. Things like walking lunges, jumping lunges and single-leg squats all build core strength and stability in a movement pattern that simulates what the body goes through when running. Learn to control these movements during the exercises and it's directly transferable to running. Running is a series of single-legged squats with a flight phase - slow it down and it looks a lot like jumping lunges. Who would've thought?
The focus on movement specificity is what develops strength efficiency. We turn our base strength into targeted strength that enables the running-specific stabilisers to be recruited more quickly and efficiently.
FLEXIBILITY I'm always cautious with this word. I'm not a big believer in being flexible "just because". You simply need to have the required range of motion to optimally perform an action. For example, lots of us have tight hips from sitting at a desk all day, and this tightness often means that we can't fully extend our legs at the hips, which in turn restricts our running gait to a shortened action. Shortened action means less output. "Anatomy for Runners" is an excellent book that enables you to identify particular inflexibilities that are restricting your movement, and provides exercises and stretches to remediate the issues. I'll repeat it: flexibility is simply about providing your joints with the ability to achieve optimal range of motion through the running action.
Combine these three components successfully, and what you develop is a mobile and stable core structure which is able to efficiently execute the action of running.
Kind-a neat really.