A batched sparse (BATS) code provides a novel two-stage coding structure for the multi-hop network, in which the outer code performed at the source node generates a potentially unlimited number of batches and the inner code at the intermediate network nodes applies network coding on packets belonging to the same batch. Previous works have studied the performance of BATS codes in the erasure channels, in which the packet loss rate \varepsilon is always assumed to be a constant on each link. However, in some application scenarios such as the Industrial Internet of Things (IIoTs) where there are a number of mobile nodes in the networks, the channel conditions could be time-variant due to the mobility of nodes, resulting the packet loss rate \varepsilon varying over time as well. Therefore this paper studies the rank distribution which is one of the most significant performance metric of BATS codes under time-variant channels by assuming the packet loss between links modeled as a random variable instead of a constant value. Closed-form expressions of rank distribution are obtained with the packet loss rate \varepsilon following two typical types of distributions. Both numerical and simulation results are provided to verify our analysis.