Do you keep your progress after a world transition?

Yes, all your permanent capital follows you through the transition. A world merge changes the map and your neighbors, not your underlying progression.

You keep:

• Your dragons, their Star Rank and their Habits
• Your Gold, your relics, Breedmarks, Cores, keys
• Your unlocked Heirlooms and their Mastery bonus
• Your alliance (it persists from one reign to the next)

You start from scratch on:

• Territory and tile ownership (new map)
• Your buildings, research and the level of your dragons (as at every reign reset)
• Your House Level (to push back up to switch the Heirlooms back on)

So: no panic when the transition comes. It's the same reign reset, simply combined with a map change caused by the server merge.