The Gemara discusses kilayim when a person and animal perform a join task. Tosfos says this is referring to owner who is pulling the rope (of the reigns) together with the animal. Since the rope is ‘batel’ to the animal and is not considered a load (even for the owner holding on to it) with regards to Shabbos, why is it considered carrying when it comes to kilayim?
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While yes, the reins are like clothing of the animal and are not a load, they aren’t really Batul. Now, Tosafos entertained the idea that simply walking with the animals would be Klayim. Although Tosafos rejected that, it shouldn’t be that hard to understand that the next level, holding the same rope, would indeed be Klayim.