Sort of. I would love for this question to be properly answered. People say it helps injuries, but after an injury in sport you want to immediately apply cold. It helps reduce swelling and stop bleeding. Even if you don’t get an injury, it’s still recomended you dive in a pool or sit in an ice bath after a big work out or a sports game to help reduce injuries. The only positives I can see from a warm down is it gets the blood flowing again and who knows maybe swelling is a good thing for PE.
Heat will also cause the ligaments to become more liquidy again which in my minds means they are more likely to retract a little to their pre-workout state. I get in the shower afterwords and make it warm, not hot and let it hit my unit for a bit. Then I give it a light stretch straight out for a bit and over about a minute turn the water down until it’s noticeably cold. This way I feel like the initial warmth gets the blood going and helps clear all the excess blood left over from your work out (this will help reduce discoloration). Then the stretch while I turn it down a little will help cement the ligaments.
It’s probably just superstition. Stats show that people will gain whether they warm down or don’t. I havn’t heard any stats or convincing evidence about reduced injuries from warm downs or any stats on the effects of a “cool down”