I own and use both the 70-300L and 100-400L. Never had a 70-200 of any form as to me, they're fitting a different niche than my interests. For practical purposes the 70-300L and 100-400L are near enough the same on their long ends, and if you get either that's where you'll probably be spending much of your time. Speed of handling on the 100-400L is great thanks to the push-pull. Older IS system often not that significant if you're shooting a moving target with it, and would need a faster shutter speed anyway to prevent motion blur. Against it, there is a bit more weight than the 70-300L. On that note, the 70-300L does have the newer IS system which can help if you're shooting static subjects, is supposed to have better sealing, and is smaller and lighter. As such I find myself taking this more if I can only take one small(er) bag on travels.
In summary, decide what range you want to use, and pick accordingly. While I haven't used the 70-200 specifically, I have also got the Sigma 120-300 f2.8. For those times I want to change range on it with teleconverters, it isn't a fun thing to do in the field. So this may apply to the 70-200s also, in that is may be best to have what you think you need in one range than to mess around. If you find the 100-400L a bit heavy, use the 120-300 for a bit. The 100-400L suddenly becomes a lot lighter!