Statistical moments of particle multiplicities in heavy-ion-collision experiments are an important probe in the exploration of the phase diagram of strongly interacting matter and, particularly, in the search for the QCD critical endpoint. To interpret experimental measures of these moments it is necessary to understand the role of experimental limitations, as well as background contributions, providing expectations on how critical behavior should be affected. We here present a framework for calculating moments of particle multiplicities in the presence of correlations of both critical and spurious origins. We also include effects from resonance decay and a limited acceptance window, as well as detector efficiency. Although we focus on second-order moments, for simplicity, an extension to higher-order moments is straightforward. We also present a brief discussion of possibilities to search for a first-order line.