The technique/framework mentioned by Jim Conant and used by Berestovskii and Plaut goes back to the paper
- J. Krasinkiewicz and P. Minc, Generalized paths and pointed 1-movability, Fundamenta Mathematicae 104 (1979), 141-153, doi:10.4064/fm-104-2-141-153.
For an update on this subject see Rips complexes and covers in the uniform category, and the last section in Steenrod homotopy (which includes a simplified proof of the Krasinkiewicz-Minc result).
Edit: I was writing this in rush for an airplane, and could not elaborate on what the references are about. Now that strong shape (and even related things like Cech cohomology and Steenrod-Sitnikov homology) have been mentioned by others this simplifies my job.
What Krasinkiewicz and Minc were doing in that paper is essentially paths in the sense of strong shape. (They don't explicitly speak of "strong shape", but on the other hand it happens that papers and books that originally developed strong shape, including Tim Porter's, and most of subsequent literature under the "strong shape" brand has been incredibly focused on either categorical or general-topology aspects and didn't care to pursue any specific geometric problems, so if you're interested in any kind of substantial results on paths in the sense of strong shape, you have to look for them elsewhere!)
In the above-mentioned paper, Krasinkiewicz and Minc proved the following wonderful theorem: If $X$ is a connected (metrizable) compactum that is disconnected in the sense of strong shape (that is, not all strong shape morphisms from a point into $X$ are the same) then there exist distinct strong shape morphisms (in fact, uncountably many ones) from a point into $X$ that are represented by genuine points in $X$. This may sound like it should be either trivial or wrong, but no, it's a deep geometric result.
