Networks of fixed bed reactors with periodically switched inlet and outlet sections are studied with reference to equilibrium limited reactions. The methanol synthesis is selected as representative reaction example and the mechanisms governing the emergence of periodic regimes corresponding to trains of traveling thermal waves are analyzed. Analytical approximations, accounting for the influence of the implemented switching strategy and in satisfactory agreement with numerical simulation, are derived for the switch time stability limits in terms of temperature fronts velocities. Large and sudden variations in the switch time stability range of any wave train solution are demonstrated to occur when the sum of the velocities of declining temperature fronts exceeds the sum of the velocities of rising temperature fronts. The illustrated results provide indications on how to design and operate the network so as to generate thermal wave trains with desired number of waves and stability limits.