At the same shutter speed, the same sized print (or downsize of a bigger image) would NOT be different no matter the density (22MP vs 36MP).
True, in order to benefit even more from a higher density sensor, it is advisable to use higher shutter speeds in such fast action situations.
Again, a 36MP sensor is no worse recording image than a 22MP at the same shutter speed, it can only be better.
Here's a scenario for you that think they're equal in noise. When you increase pixels you also need to shorten your shutter speed to not show motion blur. I tried the D800 and I had to have at least 1/2500s to freeze a bike messenger I tried outside the shop where I work, that means going to 3200 iso whereas with the 5d3 I had iso 1250 and around 1/1000s, no motion blur on either, but I was over a full stop higher in iso , making the image, of course, more noisy and less DR (that some of you are EXTREMELY conserned about).