Time-domain current measurements are widely used to characterize semiconductor material properties, such as carrier mobility, doping concentration, carrier lifetime, and the static dielectric constant. It is therefore critical that these measurements be theoretically understood if they are to be successfully applied to assess the properties of materials and devices. In this paper, we derive generalized relations for describing current-density transients in planar semiconductor devices at uniform temperature. By spatially averaging the charge densities inside the semiconductor, we are able to provide a rigorous, straightforward, and experimentally relevant way to interpret these measurements. The formalism details several subtle aspects of current transients, including how the electrode charge relates to applied bias and internal space charge, how the displacement current can alter the apparent free-carrier current, and how to understand the integral of a charge-extraction transient. We also demonstrate how the formalism can be employed to derive the current transients arising from simple physical models, like those used to describe charge extraction by linearly increasing voltage (CELIV) and time-of-flight experiments. In doing so, we find that there is a nonintuitive factor-of-2 reduction in the apparent free-carrier concentration that can be easily missed, for example, in the application of charge-extraction models. Finally, to validate our theory and better understand the different current contributions, we perform a full time-domain drift-diffusion simulation of a CELIV trace and compare the results to our formalism. As expected, our analytic equations match precisely with the numerical solutions to the drift-diffusion, Poisson, and continuity equations. Thus, overall, our formalism provides a straightforward and general way to think about how the internal space-charge distribution, the electrode charge, and the externally applied bias translate into a measured current transient in a planar semiconductor device.