In this paper, we develop a mathematical framework for modeling the time-variant stochastic channels of diffusive mobile MC systems. In particular, we consider a diffusive mobile MC system consisting of a pair of transmitter and receiver nano-machines suspended in a fluid medium with a uniform bulk flow, where we assume that either the transmitter, or the receiver, or both are mobile and we model the mobility by Brownian motion. The transmitter and receiver nano-machines exchange information via diffusive signaling molecules. Due to the random movements of the transmitter and receiver nano-machines, the statistics of the channel impulse response (CIR) change over time. We derive closed-form expressions for the mean, the autocorrelation function (ACF), the cumulative distribution function (CDF), and the probability density function (PDF) of the time-variant CIR. Exploiting the ACF, we define the coherence time of the time-variant MC channel as a metric for characterization of the variations of the CIR. The derived CDF is employed for calculation of the outage probability of the system. We also show that under certain conditions, the PDF of the CIR can be accurately approximated by a Log-normal distribution. Based on this approximation, we derive a simple model for outdated channel state information (CSI). Moreover, we derive an analytical expression for evaluation of the expected error probability of a simple detector for the considered MC system. In order to investigate the impact of CIR decorrelation over time, we compare the performances of a detector with perfect CSI knowledge and a detector with outdated CSI knowledge. The accuracy of the proposed analytical expressions is verified via particle-based simulation of the Brownian motion.