In this paper, the second-order statistics of the instantaneous mutual information are studied, in time-varying Rayleigh fading channels, assuming general non-isotropic scattering environments. Specifically, first the autocorrelation function, correlation coefficient, level crossing rate, and the average outage duration of the instantaneous mutual information are investigated in single-input single-output (SISO) systems. Closed-form exact expressions are derived, as well as accurate approximations in low- and high-SNR regimes. Then, the results are extended to multiple-input single-output and single-input multiple-output systems, as well as multiple-input multiple-output systems with orthogonal space-time block code transmission. Monte Carlo simulations are provided to verify the accuracy of the analytical results. The results shed more light on the dynamic behavior of the instantaneous mutual information in mobile fading channels.