We study the response of inertial and uniformly accelerated Unruh-DeWitt detectors in the Minkowski vacuum when they are coupled to the quantum field for a finite time interval. A finite-time detector will respond even on an inertial trajectory due to transient effects. Also, the response will depend on the manner in which the detector is switched on and off. We study the response for smooth as well as abrupt switching of the detector. The detectors are switched on and off with window functions whose width, T, determines the effective timescale for which the detector is coupled to the field. We work out in detail the response of inertial and uniformly accelerated detectors for Gaussian, exponential and rectangular window functions and also obtain a general formula for the response of these detectors when a window function is specified. The T → 0 and T → ∞ limits are discussed in detail and several subtleties in the limiting procedure are clarified. © 1996 IOP Publishing Ltd.