How to find a certain frequency in a signal that varies in time?
If I understood it right, you intend to figure out how some frequency content of a given signal evolves in time.

Next to short time Fast Fourier Transform, you have the opportunity to make use of a time-frequency analysis. I would suggest you to use either the Hilbert Transform (https://en.wikipedia.org/wiki/Hilbert_transform) or the Teager-Kaiser energy operator (http://folk.uio.no/eivindkv/ek-thesis-2003-05-12-final-2.pdf). Both methods are able to make estimation of istantaneous magnitude and frequency of a signal over time. I assume that your signal is multicomponent, therefore you should first isolate the frequency components you want to analyze by means e.g. of a filter-bank and then apply one of the two mentioned methods.
Otherwise, you could directly go for an Empirical Mode Decomposition (https://en.wikipedia.org/wiki/Hilbert%E2%80%93Huang_transform). Such a method will decompose the signal in terms of frequency components: after that you can again apply Hilbert or Teager-Kaiser methods to check how your target-frequency develops over time.