In conflict tasks, such as the Simon, Eriksen flanker, or Stroop task, the congruency effect is often reduced after an incongruent compared to a congruent trial: the congruency sequence effect (CSE). It was suggested that the CSE may reflect increased processing of task-relevant information and/or suppression of task-irrelevant information after experiencing an incongruent relative to a congruent trial. In the present study, we contribute to this discussion by applying the Diffusion Model for Conflict tasks (DMC) framework in the context of CSEs to flanker and Simon tasks. We argue that DMC independently models the task-relevant and task-irrelevant information and thus is a first good candidate for disentangling their unique contributions. As a first approach, we fitted DMC conjointly or separately to previously congruent or incongruent trials, using four empirical flanker and two Simon data sets. For the flanker task, we fitted the classical DMC version. For the Simon task, we fitted a generalized DMC version which allows the task-irrelevant information to undershoot when swinging back to zero. After considering the model fits, we present a second approach, where we implemented a cognitive control mechanism to simulate the influence of increased processing of task-relevant information or increased suppression of task-irrelevant information. Both approaches demonstrate that the suppression of task-irrelevant information is essential to create the typical CSE pattern. Increased processing of task-relevant information, however, could rarely describe the CSE accurately.