The parametric correlations of the transmission eigenvalues T i of a N -channel quantum scatterer are calculated assuming two different Brownian motion ensembles. The first one is the original ensemble introduced by Dyson and assumes an isotropic diffusion for the S-matrix. We derive the corresponding Fokker-Planck equation for the transmission eigenvalues, which can be mapped for the unitary case onto an exactly solvable problem of N noninteracting fermions in one dimension with imaginary time. We recover for the T i the same universal parametric correlation than the ones recently obtained for the energy levels, within certain limits. As an application, we consider transmission through two chaotic cavities weakly coupled by a n-channel point contact when a magnetic field is applied. The S-matrix of each chaotic cavity is assumed to belong to the Dyson circular unitary ensemble (CUE) and one has a 2× CUE → one CUE crossover when n increases. We calculate all types of correlation functions for the transmission eigenvalues T i and we get exact finite N results for the averaged conductance g and its variance δg 2 , as a function of the parameter n. The second Brownian motion ensemble assumes for the transfer matrix M an isotropic diffusion yielded by a multiplicative combination law. This model is known to describe a disordered wire of length L and gives another Fokker-Planck equation which describes the L-dependence of the T i . An exact solution of this equation in the unitary case has recently been obtained by Beenakker and Rejaei, which gives their L-dependent joint probability distribution. Using this result, we show how to calculate all types of correlation functions, for arbitrary L and N . This allows us to get an integral expression for the average conductance which coincides in the limit N → ∞ with the microscopic non linear σ-model results obtained by Zirnbauer et al, establishing the equivalence of the two approaches. We review the qualitative differences between transmission through two weakly coupled quantum dots and through a disordered line and we discuss the mathematical analogies between the Fokker-Planck equations of the two Brownian 1 motion models.02. 45, 72.10B, 72.15R