By the late 19th / early 20th century "drawing the feet" came to be contextualized with respect to the type of leather (natural; processed; synthetic) used to construct the shoe / boot. In particular, parallels were drawn between certain tanning methods and a tendency to induce or promote "drawing the feet."
FORTUNATE FEET are those that enjoy the comfortable ease afforded by the "Walk-Over" shoe. Cheap leathers "draw" the feet and make them sore. As "Walk-Over" shoes never do this, the argument seems obvious. There is nothing cheap about the "Walk-Over" except its price. Material, Style, Inside and Outside Finish and every little detail in the manufacture suggests only high-grade work. Slip into a pair of "Walk-Over" shoes and learn the true definition comfort and durability. Every size and every style for every shaped foot.
The Times, Clay Center, Kansas, Thursday, December 5, 1907
Soft skins and flexible soles help, but it is perfect tanning that makes the WALK-OVER the ideal shoe for summer wear. Shoes made from leather of a common tannage "draw" the feet. It's the heat acting upon the chemicals used in tanning. A pair of WALK-OVER Summer Oxfords will insure perfect foot comfort.
The Oregon mist. (St. Helens, Columbia County, Or.), June 25, 1909
Ukiah Dispatch Democrat, Ukiah, California, Friday, April 15, 1910
Hide and Leather (trade newspaper), Chicago, January 1, 1916
Note that the second quote above mentions heat acting upon the chemicals used in tanning. Heat apparently isn't the problem; it's part of the cause. The separation of "drawing" and heated feet seems clear in this 1935 patent application:
Actual tests of the material ... have demonstrated that my artificial leather composition is comfortable to the substantially respirable and waterproof, prevents the entry of moisture through the shoe and yet does not draw the feet nor cause the feet to sweat or heat.
PATENT OFFICE 2,146,771 I ARTIFICIAL LEATHER COMPOSITION
Horace A. Sheesley,
Tufide Products Corporation, a corporation of Maine Application
October so, 1935, Serial No. 47,512
Patented Feb. 14, 1939