At the moment, we cannot say that they are related due to Nintendo not officially releasing Volcanion, Hoopa and Dianice.
Although there are certain similarities to suggest that Carbink and Dianice could be closely related, these include:
Type- Although it is unconfirmed, it is rumoured that Carbink and Dianice both share the same type. This being Rock/ Fairy respectively.
Also look at the similarities is this image:
It is completely speculation at this point, as many will argue that Carbink looks like a baby Pokémon similar to Phione and seems like it has a further evolutionary stage. Others say that Carbink and Dianice are completely separate Pokémon and not related to each other what so ever, despite both of them having such a rare typing in the game.
According to Serebii Carbink is known as the "Jewel Pokémon" while according to Serebii and here Diancie is also known as the "Jewel Pokémon."
I don't want to be the one that says Carbink is a relation to Diancie because it's still unconfirmed, but what I will say though is that there is a lot of evidence to suggest that they are close related similar to Phione and Manaphy.
Edit: Dianicie was officially released by Pokémon on the 12th of February 2014. On the Pokémon website it was confirmed Diancie was indeed a Rock/Fairy type and will star in the new and upcoming movie "Cocoon of Destruction" now entitled "The Cocoon of Destruction and Diancie." It was also confirmed to be related to Carbink:
"Diancie is an extremely rare Pokemon, created when Carbink undergoes a strange mutation."
"Groups of Carbink tend to gather around Diancie, but Diancie pays them little heed and acts only according to its own will."
Hope I helped :)