Measurement invariance (MI) is an essential part of validity evidence concerned with ensuring that tests function similarly across groups, contexts, and time. Most evaluations of MI involve multigroup confirmatory factor analyses (MGCFA) that assume simple structure. However, recent research has shown that constraining non-target indicators to zero when cross-loadings are present results in biased estimates of latent variable associations. Using Monte Carlo simulation, we investigate the behavior of fit statistics for identifying non-invariance when the target measurement model is the same for both groups, and the source of non-invariance is the presence and magnitude of non-zero cross-loadings in one group, but not in another. We consider differences between MGCFA and multigroup ESEM (MGESEM), and combined and separate group tests of configural invariance. Implications for applied researchers are provided.