Honestly, if I could avoid using a dslr for sport I would. It's one of those situations where a smaller sensor is an advantage. I own an a7s though so, for my personal stuff, thats been the camera available.
I previously shot on the 7D which was built like a tank. By all accounts the 5D III is better again. The a7s isn't as good. That's not to say it's poorly made, but if I was filming something like skiing where the environment is quite harsh, and there was the chance of dropping it in snow etc, I'd think twice. That said, Phillip Bloom has been using the a7s all over the world recently in humid and dusty conditions and it appears to have stood up to the challenge. The other thing I would recommend is getting the a7s with the movcam cage http://cvp.com/index.php?t=product/movcam_303-2201&gclid=CMKuuIfazMICFQj..., which is built really well and would protect the camera in a fall, but obviously not help the weather sealing.
It all depends on which sports you are shooting, when and where you a shooting them and what style you want to shoot them in. All three of the cameras really good and its very hard to make a decision. What clinched my choice was having a look at shooting I admired and then working out which camera would allow me to achieve that most easily. If you love the images your getting back from the camera and what you can achieve, you'll find yourself really happy to work around the disadvantages.
Having shot a bit of sport with the a7s I can confirm the rolling shutter issue is a real issue, though it is lessened significantly in APS-C crop mode. According to this test http://www.cinema5d.com/rolling-shutter-sony-a7s-vs-arri-amira-canon-c300-5d-mark-iii-1dc-panasonic-gh4/ in APS-C the a7s actually has less rolling shutter than the 5D III. If your willing to fiddle about switching in and out of crop mode when you're on a long lens it could be ok. Not great but ok.
The reason for the comment is I think It's still worth considering. The video quality is superior to the 5D III and the high iso is incredibly useful for spot shooting as you can shoot at f8 - f11 in a poorly lit hall and get fairly clean images back.
As always, test before you buy. Talk to your local rental place. After christmas can be a great time to get a cheap hire rate as cameras aren't usually so busy.