Will Equestria ever turn into a sort of republic?
Perhaps sometime after the time period featured in the series My Little Pony Friendship is Magic, the Equestrian political system will convert into a type of republic.
Most legislative and political power would be vested in an Elected Senate, with senators elected once every four or five years. The role of a collective head of state would be vested a Ruling Triumvirate selected once a year by random lottery from the ranks of the Elected Senate. The three members of the Triumvirate would have veto power over any bill passed by a Senate majority. The executive power would be vested in a Prime Minister elected by a Senate majority.
Princess Celestia and Princess Luna would have their political power limited to being the local co-rulers of Canterlot, Princess Cadance would have her personal power limited to the borders of the Crystal Empire, and Twilight Sparkle as a princess would be a mere social and community mediator without ANY real political authority.
The capital of Equestria would be moved from Canterlot to (maybe) Ponyville.
The royal guards might have their activities limited to Canterlot and the Crystal Empire, or they might be incorporated into a new military and police force called the Equestrian National Guard.