We present a theoretical analysis of correlation properties of the local density of states in a disordered emitter probed by resonant tunnelling through a localized impurity state. The emitter is considered to be a cylinder of length L and radius R with elastic mean free path l much less than {L, R} and the effective dimensionality d less than or equal to 3 of the emitter is determined by the relation between the typical scale over which diffusion occurs, namely the quasi-particle relaxation length L-c and the dimensions L and R. The differential conductance measured in asymmetric double-barrier structures has been used (see, e.g., Schmidt T, Haug R J, Fal'ko V I, von Klitzing K, Forster A and Luth H 1996 Europhys. Lett. 36 61) to image local density-of-states fluctuations. We give analytic expressions for the variance and for correlations of the differential conductance with respect to voltage and applied magnetic field for the limits of a bulk three-dimensional emitter, a film, a wire and a pillar, and we determine the effect of magnetic anisotropy in lower dimensions. A numerical calculation, valid for arbitrary L, is performed in order to describe the crossovers between these limits where the correlation functions are sensitive to the shape of the emitter and the position of the resonant impurity.