The gem on this opening has various meanings.
The gem that the dead characters let go at the beginning is the wishes and hopes that they had while they were alive, that they want to keep alive on the world, deepened on the earth of the living. Ideals of peace, honor, freedom and justice that if it not was for their great work sharing them to others they will be lost to the time.
The gem that Naruto searches is happiness, his dreams, that he restless tries to catch through the series. The gem at the beginning for Sasuke was his dreams, of killing his brother and knowing that now that he has achieved his goal, be happy, then it changes the meaning, he is fighting with it, he has achieved his goal, but now he understands his brother and wants to actively fight against his dream, and Sakura is an intermediate on both sides, the light of both irradiate on her, she is the one that has always stood up from them, and even as her tears falls seeing them on their worst, she still want both of them alive and safe.
The transition shows Sakura being changed by Rin and Naruto and Sasuke by Kakashi and Obito, the gem that has been keeping them together -Rin- is destroyed, then Kakashi face backwards to the past, while Obito face forwards, to the future, with Madara’s plan on his head, finally Kakashi face forwards and Obito backwards, when Tobi’s identity is revealed and his motivations exposed, one is looking to the future, the other is looking to the past, then both are back to back, unknowns to each other, wanting to subsane their mistakes, finally they are both facing backwards, remembering the past and the gem that their lost appears as a reminiscent between them, then both face forwards, to the future, as they realize that that was what Rin would surely want for them, to both fight together so the pain that they felt would never been felt anymore from anyone.
Madara’s gem are his ambitions, that Kaguya took from his, as she is quietly revealed, showing that Madara’s ambitions werent his, but Kaguya’s ambitions, and that gem rightfully is her gem.
At the end we see everyone trying to catch a gem, is the gem of the peace, the ideals of peace, honor, freedom and justice that the lost ones gave to us, a torch-passing moment, also connecting with the Kakashi and Obito part, everyone tries to catch that gem, because with those ideals, nobody would have died, without war, hate, dishonor and distrust, nothing would have happened, and they need to catch it to end them all, but nobody can, even as they try, and the gem slips up from their hands, and keep falling, nobody can catch it, because nobody is free from those errors, everyone played their part on keep the cycle of blood around.
Finally, Naruto catch it surely and firmly with his hand, he get those values, and he ends the cycle of blood that has been perpetuated through the years on the Ninja world, he is the choosen to get that gem from the dead, so nobody would have their same fate again, with their lives ended early from the war and violence of the world. He also with this catches his dreams and happiness that he has been trying to find all along.
Storyboarder and director Hiroyuki Yamashita, with this, show us that he understands the series and its message better than anyone.