Transient phenomena that impact the speed of electron devices fabricated from organic semiconductor materials are of critical importance for the development of this new technology. Here we investigate theoretically and experimentally the effects of traps on the establishment and the depletion of the conducting channel in organic semiconductor field effect transistors (OFETs). The device structures explored resemble typical organic thin-film transistors with one of the channel contacts removed. The channel length is varied, and generally is longer than in typical OFETs, in order to vary the carrier transit time. By measuring the displacement current associated with charging and discharging of the channel in these capacitors, transient effects on the carrier transport in organic semiconductors may be studied. When carriers are injected into the device, a conducting channel is established with traps that are initially empty. Gradual 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 of the formation of the conducting channel and the resulting displacement currents are presented. Two simple scenarios are considered first: the spatially-averaged carrier density model neglects carrier dynamics but shows the salient trapping/detrapping effects. Second: a simple geometric model neglects traps but elucidates the effect of carrier dynamics and is used to define an effective mobility that can be extracted from experiments. Finally, numerical simulations are used to study the dynamical and trap effects together. Comparing the average trapping time with the time scale of the current transient sheds light on the influence of traps on the effective mobility. If the average trapping time is very long the effect of traps is negligible and the mobility is largely unaffected by traps on the timescale of the experiment. If the average trapping time is short the effective mobility is reduced but it is time-independent. However, the effective mobility changes with time if the average trapping time is comparable to the current transient interval, eventually reaching a constant value once the trapping and detrapping events balance. Furthermore, the temperature dependence of the effective mobility arising from the temperature dependence of the trap emission rate is explored. Experimental data for pentacene based devices of different channel length and obtained over in the temperature range 120K < T < 300K support the concepts of the physical model.