This April 1937 advert contains a trove of interesting pieces of information:
This derailleur seems to be operated by a rigid cable that can transmit torque, a bit like the ones later used by Huret to drive its speedometers. This cable appears to be twisted by some unseen control lever to turn a cog that is mounted on its end which, in turn, drives a toothed, sliding, shaft in and out. The pulley cage seems to be mounted on the end of this shaft.
The relevant patent is French Patent # 813,241 which includes some clearer drawings.
(Source gallica.bnf.fr/ Bibliothèque nationale de France)