Currently the phenomena of neutrino and neutral kaon oscillations, which take place at finite space-time intervals, are usually described in the framework of approaches that are inconsistent or not convenient for describing such processes. We develop a new QFT approach, adjusting the standard S-matrix formalism for the case. One applies the usual Feynman rules in the coordinate representation to construct the amplitude of the process and then passes to the momentum representation in a way which reflects the experimental setting. Effectively, it leads to the Feynman propagator in the momentum representation being modified, while all the other Feynman rules in this representation are kept intact. Wave packets are not employed, we work in the plane wave approximation, which simplifies the calculations considerably. The novel approach allows one to consistently describe unstable particle displaced decay and particle oscillation processes and to successfully reproduce the well-known results.