This article deals with the dynamic response of laterally unbounded, horizontally layered plates subjected to dynamic sources applied at arbitrary locations, which is ultimately a classical problem. This response is most often obtained via a modal superposition in terms of the complex Lamb modes by casting the equations in the frequency-space domain, followed by a Fourier inversion into the space-time domain. Then again, a much less often used alternative method relies on formulating the problem directly in the time domain in terms of a modal superposition in the wavenumber domain, which is followed by a Fourier inversion into the space domain, as considered in further detail herein. This alternative can offer powerful advantages in some cases, such as dealing easily with anisotropy, or with slowly propagating waves even in the absence of damping. At the same time, however, it is beset by difficulties associated with the Fourier inversion into the space domain. As a matter of fact, during forcing, the truncation of the modal series and of the numerical integrals is hindered by poorly convergent behaviors. Here we overcome both of these difficulties by considering the asymptotic, static behavior of the integrands. We find that by introducing a regularizing term, the response can be effectively separated into near and far fields, despite the fact that these frequency-domain concepts are alien to a wavenumber-time formulation. The so-defined far field is free of sharp variations and can then be computed in a numerical grid that is optimized regarding the propagating wavelengths, which only depend on the time-spectral content of the excitation and not on its space-spectrum. Finally, we also propose a hybrid way to compute the remaining near field by combining with a non-modal formulation, expressed in the wavenumber-Laplace domain.