The field theory of open quantum systems already has ample applications in physics with non-flat backgrounds, e.g. for inflation, black holes or quantum experiments in space. Open quantum systems are commonly described by reduced density matrices, which are obtained by tracing out the environmental degrees of freedom, and whose evolution is given by quantum master equations. Solving such equations often poses an intricate or even impossible task. In this talk I am going to present a practicable formalism which allows for the direct computation of reduced density matrix elements without having to consider a master equation. It is based on techniques from non-equilibrium QFT like thermo field dynamics, the Schwinger-Keldyh formalism and the Feynman-Vernon influence functional. The presented results are an early step in a research program that will unravel even more intimate relations between QFT of open systems and QFT in curved spacetimes.