Transport properties of open chaotic ballistic systems and their statistics can be expressed in terms of the scattering matrix connecting incoming and outgoing wave functions. Here we calculate the dependence of correlation functions of arbitrarily many pairs of scattering matrices at different energies on the Ehrenfest time using trajectory-based semiclassical methods. This enables us to verify the prediction from effective random-matrix theory that one part of the correlation function obtains an exponential damping depending on the Ehrenfest time, while also allowing us to obtain the additional contribution that arises from bands of always correlated trajectories. The resulting Ehrenfest-time dependence, responsible, e. g., for secondary gaps in the density of states of Andreev billiards, can also be seen to have strong effects on other transport quantities, such as the distribution of delay times.