The Inquisition wasn't even founded on true Chantry authority. The Chantry openly declared the Inquisition heretics. It was founded on the authority of Divine dead and gone. Chantry "authority" is tenuous at best. It grew both without support or authority of -any- government or faction. The new divine gives it's support to the Inquisition because she was a member of that Inquisition. She couldn't stop it even if she wanted to. Withdrawing Chantry support would change little, depending on how the Divine in question uses her authority. Always someone willing to fund the opposition.
As for the funding to continue to pour in, well that depends on what the inquisition will shift their focus to. I believe as long as Orlais can make use of their arrangement of mutual support the nobles will fund whatever is in vogue to fund.
I guess my point is that the Chantry and the Inquisition are no longer one in the same. Spun the right way you might even say the Inquisition controls the Chantry, not the other way around.
I realize this is some pages back now and probably way off topic, sorry lol. I get distracted by this work thing while I'm at work.
What she said.
I see the Inquisition as the new form of the Chantry, with the bureaucracy and clergy of the old Chantry dancing to the Inquisition's tune. Not in the Andrastian sense, but in that it is the foundation of Southern Theodosian society. It's acquired enough political capital by the end of the game to push their candidate through for the position of Divine, and the Chantry and the Orlesian Empire have thrown their hats in with them to save themselves from total collapse. Ferelden sees them as allies. They have soldiers everywhere, and everyone owes them favors. It's very easy to see the Inquisition continuing on for some years, before eventually being absorbed into the new, specialized forces for stability, like the original Inquisition being dissolved so that it could better support the Nevarran Accord. A new world order, basically. I could also, somewhat, see it persisting as a UN like force.
Of course, there is a darker parallel for a religiously motivated military organization. Namely, the Teutonic Order.
The Teutonic Order were Crusaders who decided that, rather than just invading the Levant, the Baltic was also some hot stuff. Lithuania and Prussia was full of juicy Pagans just begging to be slaughtered/converted into tax-paying citizens. While the Kingdom of Poland initially helped them out, overtime the Teutonic Order became more and more powerful, nearly annihilating the native Prussian race, and looking to expand itself into a Great Power. A panicked Poland allied itself with the Lithuanians in hopes of destroying the Order.
I can see the Inquisition being destroyed if it pushes its luck too far and upsets the big players. While it has a very powerful army, and multiple fortresses, they're spread across Ferelden, Orlais, Nevarra, and the Free Marches. Their reach is far, but if public approval fell low enough and the Inquisitor wasn't there to wield their political and religious capital, then the Inquisition could be destroyed.