We investigate the applicability of frame multipliers as compressive sensing measurements. We show that, under certain conditions, subsampled frame multipliers yield measurement matrices with desirable properties. To that end, we prove a general probabilistic nullspace property for arbitrary nonempty sets, that accounts for the special measurement structure induced by subsampled frame multipliers. Conditions for uniqueness of reconstruction of signals that are sparse with respect to dictionaries or, more generally, to non-linear locally Lipschitz mappings are obtained as special cases. Furthermore, we show that a frame multiplier matrix is full superregular, i.e., that all its minors are nonzero, for almost all frame symbol vectors, provided that the underlying frames are full spark and sufficiently redundant. Since Gabor frames are full spark for almost all windows, we study Gabor multipliers in more detail and are able to derive improved constants for some scenarios. Finally, our simulation results reveal that, in many instances, subsampled frame multiplier matrices exhibit the same ℓ1-reconstruction performance as i.i.d. Gaussian measurement matrices.