Separating out a sequence of DFTs with overlapping windows

Suppose I have a time series partitioned into $N$ equally-sized sections. Choosing some $k<N$, for each of the sections $i \in[0, N-k]$, I concatenate that section with the $k-1$ sections to the right and perform an FFT over that window. Below is a graphical representation:

With the results from all of these FFTs, can I construct the set of DFTs over each of the $N$ original sections that partition the time series without inverting each of my results?

Motivation

My underlying use case is a bit more convoluted than the case in the question, but I figured it is too specific to pose as the central focus. I am trying to reconstruct a faithful audio signal from video of a visualizer (this Reddit post details more of my process for the curious). To construct the visualizer, a DFT of the audio with a few hundred bins is collapsed into into 31 by averaging within each new bin range, then smoothed over time by widening the time-window beyond the video frame length.

Thanks to the collapsing, I have no way to recover any amount of the true original signal. However, I was hoping that I could construct the unsmoothed collapsed DFT if the solution in the uncollapsed domain is nice enough.