Traffic assignment is a core component of many urban transport planning tools. It is used to determine how traffic is distributed over a transportation network. We study the task of computing traffic assignments for public transport: Given a public transit network, a timetable, vehicle capacities and a demand (i.e. a list of passengers, each with an associated origin, destination, and departure time), the goal is to predict the resulting passenger flow and the corresponding load of each vehicle. Microscopic stochastic simulation of individual passengers is a standard, but computationally expensive approach. Briem et al. (2017) have shown that a clever adaptation of the Connection Scan Algorithm (CSA) can lead to highly efficient traffic assignment algorithms, but ignores vehicle capacities, resulting in overcrowded vehicles. Taking their work as a starting point, we here propose a new and extended model that guarantees capacity-feasible assignments and incorporates dynamic network congestion effects such as crowded vehicles, denied boarding, and dwell time delays. Moreover, we also incorporate learning and adaptation of individual passengers based on their experience with the network. Applications include studying the evolution of perceived travel times as a result of adaptation, the impact of an increase in capacity, or network effects due to changes in the timetable such as the addition or the removal of a service or a whole line. The proposed framework has been experimentally evaluated with public transport networks of Göttingen and Stuttgart (Germany). The simulation proves to be highly efficient. On a standard PC the computation of a traffic assignment takes just a few seconds per simulation day.