Since in general relativity the curvature corresponds to the gravitational field, QFT in curved spacetime corresponds to a normal QFT with an external source (=the gravitational field).
For a curvature that depends on the time, we are dealing with a time varying source that can, for example, produce particles. A proper analogy is a driven harmonic oscillator.
It is important to note that QFT in curved spacetime is not quantum gravity. In a theory of quantum gravity we would take the backreaction of quantum effects on the spactime into account. In contrast, in curved spacetime QFT the curved spacetime is merely a background structure, i.e. an external source and not treated quantum mechanically.