Experimental and theoretical studies of the angular X-ray-intensity distribution are performed at the output of microchannel plates under the condition that monochromatic radiation is incident on the microcapillary walls at grazing angles. The primary-radiation angles and energies at which a synchrotronradiation beam excites fluorescent X-rays propagating through polycapillary structures are determined. In the case of X-rays passing through microcapillaries, the fine structure of their spectra is investigated at energies corresponding to the anomalous dispersion region of the SiL 2, 3 and OK absorption edges. Theoretical calculations are carried out using a model involving a transition layer on the microchannel surface.