  We study the transition in conductance properties of chaotic mesoscopic
cavities as time-reversal symmetry is broken. We consider the Brownian motion
model for transmission eigenvalues for both types of transitions, viz.,
orthogonal-unitary and symplectic-unitary crossovers depending on the presence
or absence of spin-rotation symmetry of the electron. In both cases the
crossover is governed by a Brownian motion parameter {\tau}, which measures the
extent of time-reversal symmetry breaking. It is shown that the results
obtained correspond to the Jacobi crossover ensembles of random matrices. We
derive the level density and the correlation functions of higher orders for the
transmission eigenvalues. We also obtain the exact expressions for the average
conductance, average shot-noise power and variance of conductance, as functions
of {\tau}, for arbitrary number of modes (channels) in the two leads connected
to the cavity. Moreover, we give the asymptotic result for the variance of
shot-noise power for both the crossovers, the exact results being too long. In
the {\tau} \rightarrow 0 and {\tau} \rightarrow \infty limits the known results
for the orthogonal (or symplectic) and unitary ensembles are reproduced. In the
weak time-reversal symmetry breaking regime our results are shown to be in
agreement with the semiclassical predictions.
