The decision of whether to cross a road or wait for a car to pass, humans make frequently and effortlessly. Recently, the application of drift-diffusion models (DDMs) on pedestrians’ decision-making has proven useful in modelling crossing behaviour in pedestrian-vehicle interactions. These models consider binary decision-making as an incremental accumulation of noisy evidence over time until one of two choice thresholds (to cross or not) is reached. One open question is whether the assumption of a kinematics-dependent drift-diffusion process, which was made in previous pedestrian crossing DDMs, is justified, with DDM-parameters varying over time according to the developing traffic situation. It is currently unknown whether kinematics-dependent DDMs provide a better model fit than conventional DDMs, which are fitted per condition. Furthermore, previous DDMs have not considered reaction times for the not-crossing option. We address these issues by a novel experimental design combined with modelling. Experimentally, we use a 2-alternative-forced-choice paradigm, where participants view videos of approaching cars from a pedestrian’s perspective and respond whether they want to cross before the car or to wait until the car has passed. Using these data, we perform thorough model comparison between kinematics-dependent and condition-wise fitted DDMs. Our results demonstrate that condition-wise fitted DDMs can show better model fits than kinematics-dependent DDMs as reflected in the mean-squared-errors. The condition-wise fitted models need considerably more parameters, but in some cases still outperform kinematics-dependent DDMs in measures that penalize the parameter number (e.g., Akaike information criterion). Introducing a starting point bias provides support for the novel hypothesis of rapid early evidence build-up from the initial view of the vehicle distance. The drift rates obtained for the condition-wise fitted models align with the assumptions in the kinematics-dependent models, confirming that pedestrians’ decision processes are kinematics-dependent. However, the partial preference for condition-wise fitted models in the model selection suggests that the correct form of kinematics-dependence has not yet been identified for all DDM-parameters, indicating room for improvement of current pedestrian crossing DDMs. Developing more accurate models of human cognitive processes will likely facilitate autonomous vehicles to understand pedestrians’ intentions as well as to show unambiguous human-like behaviour in future traffic interactions with humans.