On all my rifles I carry extra ammo in a butt stock shell carrior. My favorite is one made my www.murrayleather.com.
I have ejector and extractor doubles. I usually spend more time trying to capture the brass on the ejector rifles. With the extractor rifles if I am under "pressure" , it is a simple matter to just open the rifle pluck out the empties, let them drop at your feet and reload off the shell carrier. I did this when I shot my bull elephant... However I must confess even with my buff scrabbling around about 35 or so yards away when I shot two solids, and later when at about 25 as he started to get up, and I shot another solid, reloaded and fired a 500gr Swift A Frame, I removed the fired brass from the rifle and placed in in my front pants pocket.

The reason? I just did not feel under "pressure" and felt I had plenty of time to complete the reloading process. Bottom line is that it does not make any difference to me, I can use them both, however in hunting elephants I prefer the extractor gun.
I have shot all my doubles enough that I have not had any trouble switching from one to the other.