We measure the transmission through asymmetric and reflection-symmetric chaotic microwave cavities in dependence on the number of attached waveguides. Ferrite cylinders are placed inside the cavities to break time-reversal symmetry. The phase-breaking properties of the ferrite and its range of applicability are discussed in detail. We use the random matrix theory accounting for absorption effects to calculate the universal distribution of transmission coefficients T and their energy derivatives dT/depsilon. Using the absorption strength as a fitting parameter, we find good agreement between universal transmission fluctuations predicted by the theory and the experimental data.