When the Shapiro brothers of the Slovita Press were about to run the gauntlet through rows of soldiers to carry out their sentence and flog them for the false libel, the authorities feared that the first troop had been bribed to only deliver soft perfunctory blows and so at the last minute they switched the troops, and they stripped the two brothers down bare so that the deadly blows were dealt harshly and cruely indeed! In the midst of all this, the yarmulke - the kippa or head covering fell off one of the brother's heads, he retrieved it despite the blows that continued to rain on him, as obviously anyone who ran quickly fared better, yet despite all this he sufferred it, rather than take even one step with his head uncovered. (Rav Raphael Kohen as heard from Rav Shmuel Gronem) (Shmuos VeSippurim Volume I pgs 243-244)
Another tradition is that it was the younger brother whose head covering fell off, despite being tied and bound by ropes that led him to run the gauntlet, and therefore being unable to retrieve the fallen cap, he stood still as the blows rained down upon him, he accepted the Heavenly decree upon him with love and paid no heed to the pain and suffering of the bludgeoning as the soldiers struck his immobile form, eventually someone replaced his head covering and only then did he consent to move on and miraculously he survived the ordeal with the grace of Heaven. (Seder HaDoros MiTalmidei Baal Shem Tov as cited in Likkutei Imrei Pinchos Sippurim 124)
