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Although urban transit systems (UTS) often have fixed vehicle capacity and relatively constant departure headways, they may need to accommodate dramatically fluctuating passenger demands over space and time, resulting in either excessive passenger waiting or vehicle capacity and energy waste. Therefore, on the one hand, optimal operations of UTS rely on accurate modeling of passenger queuing dynamics, which is particularly complex on a multistop transit corridor. On the other hand, capacities of transit vehicles can be made variable and adaptive to time-variant passenger demand so as to minimize energy waste, especially with the emergence of modular vehicle technologies. This paper investigates operations of a multistop transit corridor in which vehicles may have different capacities across dispatches. We specify skewed time coordinates to simplify the problem structure while incorporating traffic congestion. Then, we propose a mixed integer linear programming model that determines the optimal dynamic headways and vehicle capacities over the study time horizon to minimize the overall system cost for the transit corridor. In particular, the proposed model considers a realistic multistop first-in, first-out (MSFIFO) rule that gives the same boarding priority to passengers arriving at a station in the same time interval yet with different destinations. A customized dynamic programming (DP) algorithm is proposed to solve this model efficiently. To circumvent the rapid increase of the state space of a typical DP algorithm, we analyze the theoretical properties of the investigated problem and identify upper and lower bounds to a feasible solution. The bounds largely reduce the state space during the DP iterations and greatly improve the efficiency of the proposed DP algorithm. The state dimensions are also reduced with the MSFIFO rule such that all queues with different destinations at the same origin can be tracked with a single boarding position state variable at each stage. A hypothetical example and a real-world case study show that the proposed DP algorithm greatly outperforms a state-of-the-art commercial solver (Gurobi) in both solution quality and time.