Once you finish a Collection and trade in your Collectibles, you will receive the reward featured in the Collection. The Collectibles required to finish the Collection will be deducted from your inventory. This means that any other Collections that use the same Collectibles will also be reduced. Keep in mind that your Collectibles inventory is global, and not individual to each Collection.
 
Once the Collection is finished and claimed, a new Collection will take its place, except for Limited Collections which are not permanent and time-based. In the case of Legendary & Minion Collections, you will be given the option to choose between two new dragons (you can select the "i" icon, to see the details about the Dragons in question). Whichever dragon you pick will be the featured reward for the new Collection.