How does this work exactly?
Keep it simple,
Case coverings are thin and closer to the fire and get hot quicker than thick turbine disks. Therefore they expand quicker under the heat and open up blade tip clearances which reduce engine efficency.
The idea is to slow down the heating rate of the cases until the thick turbine rotor disks can catch up, so you meter some cooling air arround the outside of the case for a period of time to adjust the differences in themal growth.