The Fourier transform is a theoretical construct. It does not bother about periodicity. If you have a changing signal, you can view its sinuidal components as "carrier" and the envelope as "signal", and the Fourier transform will be the convolution of the transform of the carrier (usually just a pair of dirac pulses at the carrier's frequency) and the Fourier transform of the envelope.
Now the envelope usually is significant more as a time sequence than in frequency space. Its changes are also at a much lower speed than the carrier oscillation.
So it usually makes sense to do analysis using short-term Fourier transforms. Those are good at detecting the carriers at a transitory envelope value under the assumption that the envelope changes are few compared to the window length. A limited time window also limits your frequency resolution: overlaps and windowing are used for getting the most of your data.
Now high frequency carriers can reflect envelope changes a lot better than low frequency carriers (or rather: frequency differences can be a lot larger, requiring smaller times to tell different frequencies apart): that makes multi-scale analysis attractive, and then we are no longer talking about short-term Fourier transforms for the time-frequency analysis needed to pick apart frequencies and envelopes.