The correlation-corrected balanced estimation of diffusion entropy (CBEDE) was used to evaluate the local scale behavior of stock index volatility series and its evolution behavior. The volatility series of four important stock indices in China's stock market from January 9th 2014 to January 5th 2017 were investigated. It is found that all the segments with a 10 trading day length show a scale-invariant behavior. On different time scales, the evolution of scaling behavior shows rich characteristics. In particular, in the duration of important events in China's stock market, the scale behavior has different patterns, or falls then rises, or rises, or falls. These characteristics can be used to quantitatively describe financial events and the differences between them.