The paper presents the results of an analysis of the process of drawing AISI 321 stainless steel thin-walled seamless tubes on a floating plug. The influence of the geometry of dies and plugs, drawing velocity, and lubricants on the possibility of carrying out the pipe drawing process without a loss of strength of the lubricating film and, consequently, disturbance of the forming process and tube cracking, and also on the temperature in the drawing process, the mechanical properties of the tubes drawn, and the microhardness and roughness of the inner and outer surface of the tubes was investigated. The parameters of the drawing tools used were as follows: angle of drawing dies α = 16° and floating plugs with angles of inclination of the conical part of the plug β = 11.5°, 13°, and 14°. The drawing dies and floating plugs were made of G10 sintered carbide. Drawing speed was varied over the range 1 to 10 m/min. The study used several lubricants. Tubes with dimensions (outer diameter D0, wall thickness g0 before drawing process) D0 = 19 mm, g0 = 1.2 mm and D0 = 18 mm, g0 = 1.2 mm were drawn to produce tubes with dimensions (outer diameter Dk, wall thickness gk after drawing process) Dk = 16 mm, gk = 1.06 mm on a drawbench with the same total elongation, while the diameter and wall thickness were changed. During the process, continuous measurements were made of the drawing force and temperature in the deformation zone and on the tube surface. It was found that the drawing process causes a decrease in the roughness parameters Ra and Rz of the inner surface of the tubes. Moreover, after drawing, an increase of 30–70% was observed in the microhardness of the tube material in relation to the microhardness of the charge material. Based on the test results, it can be concluded that the work of frictional forces is the main direction of optimization of tube drawing on a floating plug process of hard-deforming materials.