How exactly did Obito escape from Itachi's Amaterasu? Itachi attacked Obito with Amaterasu through Sasuke's eyes, when Sasuke saw Obito's Sharingan. As soon as Obito was engulfed by the flames, he went into the dark region of the room and got rid of the lethal flame. How did he do it? Did he achieve that with Kamui?
I've been thinking about this question, and I've found a theory, which would explain a few things:

I'm almost 100% sure that the Amaterasu had hit Obito, as in the anime we can hear the damage being received. As from that moment, we can't talk about the Transportation Technique, because the Amaterasu lasts until his target is completely gone.

So that would reach the 5 minute limit and Obito would have been dead, just as it would've been the same result in the fight with Konan: those paper bombs kept coming, and they would've reach the 5 minutes limit, IF, and here it comes my theory. Obito escaped with the same technique both times, the Izanagi.

Since he had a lot of Sharingan eyes in reserve, he could have easily replaced the eye he had just used, just as he did after fighting Konan. AND he could have implied this, by the lines (if I recall well), "Even Itachi didn't know everything about me". That he could use Izanagi, or that he has those Sharingan reserves (since Itachi never knew about the eyes collected by Tobi, or Tobi's expertise in using this rare and forbidden jutsu).