What are the benefits/disadvantages of doing core exercises before/after or during a workout?
Performing core exercises before training the rest of the body has been theorized by some experts to better prepare the body, from a neurological standpoint, for "load." This load may be ANAEROBIC and/or AEROBIC. According to this method, training the appropriate muscles of the core (as determined by the assessment) prior to multi-joint/complex exercises electrically "triggers" them, essentially enabling the individual to have greater proprioception of and control over the spine and pelvis. However, the fitness/performance professional must be careful not to overly fatigue these important muscles prior to exercise as, once again, they are the spine/pelvis's support network. If totally fatigued, for example, prior to a loaded back squat and/or deadlift, poor form and injury may result.
Isolating particular muscle groups of the lumbo-pelvic hip complex after complex exercise is done to avoid the over fatigued state mentioned above. When done so correctly, training these muscles at the end of a workout can be done more aggressively with the goals of improving posture and function and without as great a need of the support for any complex exercises still to come.
Training core muscles during the workout (i.e., between sets, etc.) may be done so long as it does not compromise the quality and/or safety of any of the complex primary movements/exercises being performed. Similar to the example above, one would not want to perform a lower abdominal exercise to fatigue in between heavy sets of deadlifts as the lower abdominals are crucial for proper spinal/pelvic stabilization during this movement pattern.
Essentially, it's all about the body's ability to function and when, where and how it functions best. It's hard to generalize because different individuals and their goals may require totally different approaches. Essentially, NEURAL adaptations (i.e., functionality) should become your primary focus rather than absolute strength goals or aesthetics. Training isolated movements for aesthetics (i.e., crunches, sit ups, etc.) can lead to serious muscular imbalances that can in turn lead to visceral (i.e., internal organ) dysfunction and future disease states. Take a common sense approach.