We consider a transportation system consisting of a linear network of N + 1 terminals served by S vehicles of fixed capacity. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to some terminal j, 0 ≦ j ≦ i − 1, and are served as empty units of vehicle capacity become available at i. The vehicle fleet is partitioned into N service groups, with vehicles in the ith group stopping at terminals i, i − 1,···,0. Travel times between terminals and idle times at terminals are stochastic and are independent of the customer arrival processes. Functional central limit theorems are proved for random functions induced by processes of interest, including customer queue size processes. The results are of most interest in cases where the system is unstable. This occurs whenever, at some terminal, the rate of customer arrivals is at least as great as the rate at which vehicle capacity is made available.