These three moments are so interesting, because for once we see Thorin as the uncle, the father-figure and carer, instead of Thorin the King.
It’s first apparent when Kili gets shot. The disbelief and horror in Thorin’s whisper of 'Kili' says so much more than he’s ever usually willing to give away. He retreats back into himself when he sees that Kili has escaped with them, that he seems to be alright, considering. But in that moment, when he heard his nephew cry out in pain, Thorin was not a leader and king utterly set on his mission and willing to make necessary sacrifices; he was an uncle horrified at the sudden reality that he could lose his youngest charge. And Thorin was right up against the gate whilst Kili got shot. He couldn’t see what had happened - he only heard Kili’s cry of pain and Fili’s horrified shout. For a brief second, Thorin thought the worst had happened.
When he first stops Kili from getting on the boat in Laketown, he’s still hiding behind his kingship. ‘Not you’ is abrupt. ‘We will travel at speed, you will slow us down’ is coldly logical. But when Thorin brings himself to look properly at Kili, his entire expression softens. He can’t keep up the mask of indifferent kingship, and he can’t keep Kili at arm’s length, either; it’s Thorin who pulls Kili closer, Thorin who quietens to make their moment more private, Thorin who gives Kili a small, affectionate, caring smile and puts his hand on the back of Kili’s head to cradle him closer just for a moment. It’s a moment that has been coming ever since Thorin’s horrified whisper of Kili’s name back in the barrels.
And then he turns away and his mask goes straight back up so that the rest of the Company don’t catch the softness in his face - then, he’s met with Fili. And when Fili appeals to him for his brother’s sake, he’s not appealing to his king, but for once directly to Thorin as a member of his family. At Fili’s call of ‘Uncle…’, Thorin’s struggle to keep up his indifferent mask is obvious in his expression. He’s just allowed himself a moment to be soft and quiet and familiar with one nephew; and he’s trying hard to maintain the composure of the King with the other. He doesn’t quite manage it. It’s not his king’s grip that Fili tears himself out of to return to Kili’s side; it’s his uncle’s hand on his arm, and Thorin knows it too, because he lets Fili go.