We present device simulations exploring the effects of traps during transient processes in the conducting channel of organic field effect transistors (OFETs). The device structure explored resembles a typical organic thin-film transistor with one of the channel contacts removed. However, the channel length is much longer than in typical OFETs in order to increase the transit time. By measuring the displacement current in these long-channel capacitors, transient effects in the carrier transport in organic semiconductors may be studied. When carriers are injected into the device, a conducting channel is established while traps, which are initially empty, are being populated. The filling of the traps then modifies the transport characteristics of the injected charge carriers. In contrast, dc experiments as they are typically performed to characterize the transport properties of organic semiconductor channels investigate a steady state with traps partially filled. Numerical and approximate analytical models for the formation of the conducting channel and the resulting displacement current are discussed here. The temperature dependence of the effective mobility arising from the temperature dependence of the trap emission rate is explored, and calculated results are compared with experimental data. We show that displacement current measurements on OFET structures provide unique opportunities for the study of trap dynamics involving a wide range of time scales.