In the case of the Little Rock Nine, the 101'st airborne was deployed to protect the students until federalized Arkansas National Guard units could be deployed. In 1957 the 101st was stationed at Fort Campbell, Kentucky, and thus was the closest airborne unit to Little Rock. As part of the Cold War readiness, Airborne units tended to maintain a state of being that allowed them to deploy very quickly, and so you are correct in that they were called in primarily because they could get there very quickly. Of interest is that, due to the nature of the desegregation efforts at the time, they deployed without members of the unit who were black, who rejoined the unit after they were replaced by the national guard.

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The Ole Miss riots however did not involve airborne troops. U.S. Marshalls were backed up by elements of the 70th Army Engineer Combat Battalion, also from Fort Campbell but not related to the airborne stationed there, as well as the 503rd and 716th Military Police Battalions and a contingent of Naval medical staff from a nearby navy hospital. In this case as well however, the units were picked from what was available to deploy quickly, as both of the MP battalions were already in the area on an exercise.