In this paper, we consider a diffusive mobile molecular communication (MC) system consisting of a pair of mobile transmitter and receiver nano-machines suspended in a fluid medium, where we model the mobility of the nano-machines 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 introduce a statistical framework for characterization of the impulse response of time-variant MC channels. In particular, we derive closed-form analytical expressions for the mean and the autocorrelation function of the impulse response of the channel. Given the autocorrelation function, we define the coherence time of the time-variant MC channel as a metric that characterizes the variations of the impulse response. Furthermore, we derive an analytical expression for evaluation of the expected error probability of a simple detector for the considered system. In order to investigate the impact of CIR decorrelation over time, we compare the performances of a detector with perfect channel state information (CSI) knowledge and a detector with outdated CSI knowledge. The accuracy of the proposed analytical expression is verified via particle-based simulation of the Brownian motion.