Non-Linear, Non-Stationary spectral analysis methods! When and where?
Non-linearity and non-stationarity are non-properties. Without more details, they do not say much about the methods that may perform well, and moreover the  choice depends a lot on what you really do: analysis, feature extraction, enhancement, filtering, component separation, restoration? What follows are typical sets of tools you could use:


Your moving-window-Fourier definitely has some STFT flavors, and there are many non-linear extensions: bilinear or quadratic time–frequency distributions (Wigner-Ville, Choi–Williams, Rihaczek)
Non-linearity may come from changing the classical energy preservation into other norms, like robust DFTs (under an $\ell_1$ norm minimization), see for instance Robust $L$-estimation based forms of signal transforms and time-frequency representations, see also Using different algorithms/formulas to create spectrograms
Some methods also alleviate the hypothesis that signal formation is linear: variance-stabilizing transforms,  cepstral analysis or homomorphic signal processing (linear transforms applied in a $\log$ domain)
Generalizations of linear approaches in polynomial or rational fraction modeling, with a recent interest in Rational Optimization for Nonlinear Reconstruction with Approximate $\ell_0$ Penalization.


All of the above can be combined somehow, like EMD at different scales in Intrinsic multi-scale analysis: a multi-variate empirical mode decomposition framework. EMD has many uses and some known limitations at the same time.