The response of convectively unstable equations to a localized oscillatory forcing (i.e. the ‘signalling problem’) is studied. The full mathematical structure of this class of problems is elucidated by examining partial differential equations of second-order (the linear Ginzburg–Landau equation) and fourth-order (the linear Kuramoto–Sivashinsky equation) in space. The long-time asymptotic behaviours of the Fourier–Laplace integral solutions are obtained via contour integration and the method of steepest descent. In the process, a general algorithm is developed to extract the important physical characteristics of such problems. The algorithm allows one to determine the velocities that bound the transient and spatially growing portions of the response, as well as a closed-form transfer function that relates the oscillatory disturbance amplitude to that of the spatially growing solution. A new velocity is identified that provides the most meaningful demarcation of the two regions. The algorithm also provides a straightforward criterion for identifying ‘contributing’ saddles that determine the long-time asymptotic behaviour and ‘non-contributing’ saddles that give errant solutions. Lastly, a discontinuity that arises in the long-time asymptotic solution, identified in prior studies, is resolved.