The dynamics of an open quantum system can be fully described and tomographically reconstructed if the experimenter has complete control over the system of interest. Most real-world experiments do not fulfill this assumption, and the amount of control is restricted by the respective experimental setup. That is, the set of performable manipulations of the system is limited. In this paper, we provide a general reconstruction scheme that yields an operationally well-defined description of non-Markovian quantum dynamics for a large class of realistic experimental situations. The resultant "restricted" descriptor for the process, surprisingly, does not have the property of complete positivity. Based on these restricted dynamics, we construct witnesses for correlations and the presence of memory effects. We demonstrate the applicability of our framework for the two important cases where the set of performable operations comprises only unitary operations or projective measurements, respectively, and show that it provides a powerful tool for the description of quantum control experiments.