I've actually been experimenting with this a bit lately shooting youth soccer. I'm also using the 70-200 f2.8 II (with the 1DX), and I have the 1.4x II and 2x III extenders (actually, I have the 2x II extender too). I've been shooting with the bare lens and with these two extenders to compare the results. I often shoot at f5.6, and for comparison purposes, that's what I used for each test. One thing that surprised me was the IQ with the 2x III extender...in a good way. It was better than I expected and nearly as sharp as the 1.4x and bare lens. The main issue I had with the 2x III was focus speed. When it would lose focus, it would go off into la-la land and slowly hunt and find focus again. If I tried to focus quickly on some action, I sometimes missed focus. The resulting photos were either right on the money or a complete miss. I didn't notice the focus speed difference so much with the 1.4x II. I plan to keep experimenting because I really do like that range of 140mm-400mm for shooting 8v8 soccer!
Cameras: 1DX, 1D4, 20D, 10D, S90, G2
Lenses: Canon 10-22mm, 16-35mm f2.8L II, 24-70mm f2.8L, 70-200mm f2.8L IS, 300mm f2.8L IS, 200mm f2L IS, 50mm f1.4, 50mm f1.2L, 85mm f1.2L, 1.4x TC, 2x TC, 500D macro, Zeiss 21mm
Lighting: 580EX, Elinchrom 600 RX's, D-Lite 4's, ABR800, 74" Eli Octa, 100cm/70cm DOs, Photoflex Medium Octa and reflectors, PW's, Lastolite Hilite, Newton Di400CR bracket