The technique/framework mentioned by Jim Conant and used by Berestovskii and Plaut goes back to the paper
J. Krasinkiewicz and P. Mine, Generalized paths and pointed 1-movability
For an update on this subject see http://front.math.ucdavis.edu/0706.3937 and the last section in http://front.math.ucdavis.edu/0812.1407 (which includes a simplified proof of the Krasinkiewicz-Minc result on this subject).