We report on experimental investigation of the effect of flux compression in superconducting YBa2Cu3Ox (YBCO) films and YBCO/CMR (Colossal Magnetoresistive) multilayers. The flux compression produces positive magnetic moment (m) upon the cooling in a field from above to below the critical temperature. We found effect of compression in all measured films and multilayers. In accordance with theoretical calculations, M is proportional to applied magnetic field. The amplitude of the effect depends on the cooling rate, which suggests the inhomogeneous cooling as its origin. The positive moment is always very small, a fraction of a percent of the ideal diamagnetic response. A CMR layer in contact with HTS decreases the amplitude of the effect. The flux compression weakly depends on sample size, but sensitive to its form and topology. The positive magnetic moment does not appear in bulk samples at low rates of the cooling. Our results show that the main features of the flux compression are very different from those in Paramagnetic Meissner effect observed in bulk high temperature superconductors and Nb disks.