This work considers the problem of flow-induced diffusive molecular communication under various mobility conditions such as (i) both transmitter (TX) and receiver (RX) nanomachines are mobile, (ii) TX is mobile and RX is fixed, and (iii) TX is fixed and RX is mobile. Closed-form expressions for the probability density function (PDF) of the first hitting time under the aforementioned mobile scenarios are derived, by characterizing the movement of the nanomachines and information molecules using Brownian motion with positive drift. The derived PDF expressions are validated through particle-based simulations. Based on these results, the performance of molecular communication with on-off keying (OOK) modulation in flow-induced diffusive channels is investigated. In particular, closed-form expressions for the probabilities of detection and false alarm with optimal Likelihood ratio test (LRT) based decision rule, probability of error, and the capacity in the presence of inter-symbol interference, counting errors, and noise from the other sources are derived. Simulation results are presented to verify the theoretical results and to yield insights into the system performance for different mobility conditions.