Multiple change point detection is a current topic in research and captures problems in a wide range of fields, for example, finance, quality control, medicine or climate control. The MOSUM (moving sum) procedure investigated by Eichinger and Kirch (2016+) is one of the basic approaches to detect multiple changes in the classical mean change model. In order to generalize their results to several change point situations, MOSUM statistics based on estimating functions are used. Thereby only the global estimator of the parameter has to be computed which could be a huge advantage regarding the computational effort. After constructing an appropriate MOSUM test statistic we examine its asymptotic behavior under the null hypothesis and alternatives as well as statistical properties of corresponding estimators.