The Mongol capital should flip in any case, afaik.
Mongols haven't spawned yet so I can't argue with this from experience.

I was relying on this code in getConvertedCities in the DoC 1.16 R&F.py:
Code:
# Leoreth: remove capital locations
for city in lCities:
if city.getOwner() < iNumPlayers:
if (city.getX(), city.getY()) == Areas.getCapital(city.getOwner()) and city.isCapital():
lCities.remove(city)
However, I had already decided against using the Karakorum-as-Turk-capital gambit in my current game (I own better cities, already developed, in Persia and the ME so I decided to play it straight and save the weirdnesss for later games where I know the ropes better) but I think I'll fork my savegame just before the Mongol spawn and WB the capital to Karakorum in the fork, just to test this.
It's been a long-standing tradition in all flavours of RFC that capitals don't flip. It has supported many a startling exploit (Roman capital in London, forsooth). If DoC is abandoning this tradition (which is in any event ahistorical; RL capitals have no magical protection against fifth columns) it opens up interesting possibilities. In some cases, if the old capital is taken by invasion or flipping and the old country has become noniviable (e.g. collapsing stability) but has not yet been completely destroyed, refugees from the threatened civ could flee their country and immediately create a new, restabilised heartland elsewhere (possibly flipping cities of some other civ) instead of respawning later. Some of the hordes that threatened the Roman Empire and later Europe were fleeing fiercer hordes that were taking their old lands. When the Crusaders are driven out of Antioch or Jerusalem they could on occasion immediately flip Rhodes or Cyprus, if they don't already have those (I believe the islands are scripted respawn points already, so this is effectively just making it an instant respawn instead of delayed). The old country would cease to be core territory.
I note that the Mongols get a first contact "horde event" (conquerors stacks) against Mughals, Persia, Arabia, Byzantium and Russia but not, apparently, the Turks. Of course the Turks are going to lose half their empire to the flip anyway and then have their cities thrown into disorder at the Mongol approach, so I guess adding conquerors would have been overkill.
Edited to add (5 hours later) - I am amused by one aspect of the movement of the Turk core to Persia - that being, they can't build Settlers there initially. Not really an issue at first (after all, they need those cities building population for core population stability, not settlers) but eventually they will need more Settlers to fill out their new core. So once I had my Silk Road and my 300 Culture in Orduqent, my northern cities switched to building Settlers. And they'll keep building Settlers till the Mongols flip them, because you never know when a spare Settler might come in handy, plus it helps stability by keeping those now-non-core cities small. Still, I wonder if that was intentional (in which case, well done O Devious One) or inadvertent. Fortunately, Mirat-ul-Memalik (which is Core) and (if you capture it) Lahore (which is foreign core) can also build Settlers at need.
ETA ETA: After moving my capital to Shiraz (888 AD), suddenly I can't build Setlers in Orguqent, Kashgar, Lahore or Mirat-ul-Memalik any more but I can now build them in the core cities that were previously blocked, plus in Anatolia, Samarkand and Merv. So now the North is building Workers instead.