In Day of The Doctor, why was the 11th Doctor flying the 10th's TARDIS? They get in 10's TARDIS, but 11 starts flying it.
The 11th Doctor appeared to the 10th through a time fissure:
10th Doctor: Back, both of you, now! That's a time fissure. A tear in the fabric of reality. Anything could happen.
The 11th Doctor is the newest regeneration of the Doctor, and 10 believes that 11 should remember what's going on:
10th Doctor: Okay, you used to be me, you've done all this before. What happens next? 11th Doctor: I don't remember.
10th Doctor: How can you forget this?
11th Doctor: Hey, hang on. It's not my fault. You're obviously not paying enough attention.
When the 3 Doctors get to the Tardis it starts having problems:
10th Doctor: Ow! The desktop is glitching.
War Doctor: Three of us from different time zones. It's trying to compensate.
11th Doctor: Hey, look. The round things.
10th Doctor: I love the round things.
11th Doctor: What are the round things?
10th Doctor: No idea.
11th Doctor: Oh dear, the friction contrafibulator. Ha! There, stabilised. (The Tardis changes again.)
10th Doctor: Oh, you've redecorated. I don't like it.
11 stabilized the Tardis to his "desktop", 10 doesn't like it. It may be akin to going from one Linux distribution to another or from Windows 7 to Windows 10, everything is there but maybe not where you expect, so it made sense for 11 to pilot his "desktop".
- Eleven instigated the adventure when he met Ten through the time fissure.
- The fissure was happening to Eleven, not Ten who was a bystander .
- Eleven was the furthest point in the Doctor's time line so he should have the best chance of remembering what is occurring .
- The Tardis was set to Eleven's desktop.
Out of universe, Matt Smith was the incumbent Doctor, so he got to take the lead over David Tennant.
Plus, it does always compensate to the LATEST Doctor's incarnation (conviniently fitting with the out-of-universe reality, where the latest doctor should take the lead...)
I do like how they did give Hurt the stellar. Sort of. He was in and out-universe main character after all. He was hurting, a terrible choice thrusted upon him. And the moment was just that. A moment to make the right choice; to get help, to think it through and solve it. Anyway, that last part was just poetic, not adding much to the answer :P.