  We present a general variational principle for the dynamics of impurity
particles immersed in a quantum-mechanical medium. By working within the
Heisenberg picture and constructing approximate time-dependent impurity
operators, we can take the medium to be in any mixed state, such as a thermal
state. Our variational method is consistent with all conservation laws and, in
certain cases, it is equivalent to a finite-temperature Green's function
approach. As a demonstration of our method, we consider the dynamics of heavy
impurities that have suddenly been introduced into a Fermi gas at finite
temperature. Using approximate time-dependent impurity operators involving only
one particle-hole excitation of the Fermi sea, we find that we can successfully
model the results of recent Ramsey interference experiments on $^{40}$K atoms
in a $^6$Li Fermi gas [M.~Cetina et al., Science \textbf{354}, 96 (2016)]. We
also show that our approximation agrees well with the exact solution for the
Ramsey response of a fixed impurity at finite temperature. Our approach paves
the way for the investigation of impurities with dynamical degrees of freedom
in arbitrary quantum-mechanical mediums.
