  Classical spectral methods are subject to two fundamental limitations: they
only can account for covariance-related serial dependencies, and they require
second-order stationarity. Much attention has been devoted lately to
quantile-based spectral methods that go beyond covariance-based serial
dependence features. At the same time, covariance-based methods relaxing
stationarity into much weaker {\it local stationarity} conditions have been
developed for a variety of time-series models. Here, we are combining those two
approaches by proposing quantile-based spectral methods for locally stationary
processes. We therefore introduce a time-varying version of the copula spectra
that have been recently proposed in the literature, along with a suitable local
lag-window estimator. We propose a new definition of local {\it strict}
stationarity that allows us to handle completely general non-linear processes
without any moment assumptions, thus accommodating our quantile-based concepts
and methods. We establish a central limit theorem for the new estimators, and
illustrate the power of the proposed methodology by means of a simulation
study. Moreover, in two empirical studies (namely of the Standard \& Poor's 500
series and a temperature dataset recorded in Hohenpeissenberg) we demonstrate
that the new approach detects important variations in serial dependence
structures both across time and across quantiles. Such variations remain
completely undetected, and are actually undetectable, via classical
covariance-based spectral methods.
