Is there a name for the procedure of taking the FT over separate consecutive small time-blocks?
The Short Term Fourier Transform (STFT) 

https://en.m.wikipedia.org/wiki/Short-time_Fourier_transform

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For your question, the windows are finite and non overlapping.

The continuous time case is probably more of a mathematical device that could be approximated by diffraction over some periodic, channelized structure.

The discrete time case is very commonly used, although as RJB pointed out, usually with overlap.

If your signal consisted of flat noise and a set of sinusoids, for large DFT segments, the bins would be approximately statistically independent in time and frequency for a non overlapping window, so there are applications where overlapping is not used.