The slow progression of Kakashi's Sharingan...the long road to intangibility.
Even though Kakashi awakened his Mangekyou Sharingan at a fairly young age, courtesy of the Raikiri through Rin; he never fully learned how to activate it, as the eye manifested just before he passed out and fell to the ground; never to remember it for years to come....
But what he did use his Sharingan for, was copying jutsu masterfully. As the copy ninja became somewhat of an icon in various villages and lands, Kakashi kept copying jutsu after jutsu, fighting battles; one after the other, until he was deemed as 'Sharingan no Kakashi' the man who had copied over a 1000 jutsu.
So what was it that made Kakashi get reacquainted with his long forgotten power? In my theory, it was the visit paid by Uchiha Itachi to Konoha. The trauma that Itachi put Kakashi through, in just 3 seconds in the real world, but 3 whole days and nights in Tsukuyomi, reawakened Kakashi's jutsu by perhaps causing an effect similar to the copying abilities of the Sharingan. He already had the power hidden away, upon seeing Itachi's Mangekyou, maybe something got triggered again, thanks to this effect.
And finally....Kakashi had begun his ascent to the true power of Kamui!
Unlike the Uchiha, who are able to use this power right from the get go, Kakashi's progress was slow, but steady. As he did not have Uchiha blood coursing through his veins, Kakashi had one less thing to worry about; which was losing its light, as it is clear, that the curse of seeking new light, was only the Uchiha's to bear.
Being immune to going blind, Kakashi still had to pay the price of this curse another way. That way would prove extremely taxing for the man who did not possess much chakra to begin with, every Kamui he cast, cost him an incredible amount of stamina.
Due to this, Kakashi had to stick to warping small objects ex: Deidara's elbow, Nails etc...
But this did not mean that Kakashi was not able to warp big objects, as we have seen, when the situation called for it, Kakashi paid no mind to his own depleted health and stamina, and simply; let her rip!
Saving people's lives on more than one occasion, Deidara's blast, Pains missile. Warping bigger objects took their toll on him though, after Kakashi was forced to warp anything bigger than a kunai or a nail, he would simply, fall flat on his face; and needed to be carried thereafter.
This has been shown in his battle against Obito. As warping kunais and rasengans was easily incorporated along with brilliant battle strategy. When Kakashi had to warp Narutos entire clone however, it was all he could give; it sapped his strength and Kakashi fell to the ground.
I don't know if you've noticed, slowly but surely, Kakashi has been training his Kamui to the point where he can emulate Obito's Kamui, it's true power, which allows Obito to go in and out of their shared dimension and places far far away, in the blink of an eye; although it still needs more work. Whether he will also gain the intangibility that Obito possesses is yet unclear, but you can be sure, that Kakashi is not done progressing with his Mangekyou Sharingan, especially now that he has gotten a huge boost from Kurama and the Juubi has joined the fray.