  An ever broader range of physical platforms provides the possibility to study
and engineer quantum dynamics under continuous measurements. In many
experimental arrangements the system of interest is monitored by means of an
ancillary device, whose sole purpose is to transduce the signal from the system
to the measurement apparatus. Here, we present a method of adiabatic
elimination when the transducer consists of an arbitrary number of bosonic
modes with Gaussian dynamics while the measured object can be any quantum
system. Crucially, our approach can cope with the highly relevant case of
finite temperature of the transducer, which is not easily achieved with other
methods. We show that this approach provides a significant improvement in the
readout of superconducting qubits in circuit QED already for a few thermal
excitations, and admits to adiabatically eliminate optomechanical transducers.
