Here's some advice on tank-swapping in general that works well for me personally. For swapping in from OT to MT, what I find works best (as a PLD) is to use Fast Blade, Savage Blade (preferably used slightly before the swap is needed), then Provoke and Rage of Halone immediately after. By the time you hit Provoke, you want to be right where the other tank is/should be to avoid turning the boss (assuming the boss cleaves). When swapping from MT to OT, losing a little DPS really isn't a big deal. You should have plenty of aggro, so when the swap is very close, stop using skills. Or, when the swap mechanic hits, stop. Basically, make it so when the OT provokes, they can keep aggro even if they don't use a skill immediately. If you Rage of Halone right after they provoke, well, that's going to make the swap a little annoying. If you are better geared than the other tank, right after the swap, keep a close eye on your aggro and possibly refrain from using Rage of Halone for a rotation or two to let them get a lead on you. You may also want to swap from Shield Oath to Sword Oath. Keep in mind though when swapping back to being MT, Shield Oath cancels combos. So, don't do Fast Blade, Savage Blade, Shield Oath, Provoke, do Shield Oath, Fast Blade, Savage Blade, THEN Provoke.
At least, that's how I do tank swaps.
Many places where you need to tank swap, there's a natural time to do it during which the boss is not attacking so you don't have to worry so much about turning the boss and having it cleave the party. Titan Ex, for instance, if you're swapping, you can do so when standing underneath him for that stomp move he does. For T6 phases 1-2, you can swap when the flower appears and she's distracted with the devour move.
Garuda Ex Spiny Plume swaps are different from other swaps. With them, you never want to attack them because of how fragile they are. So, in theory, you should be able to pull the plume off the other tank by just provoking it. I usually flash too afterward just in case.
Here's a (non-complete) list of some fights where people sometimes do tank swaps (though many/most can be avoided by overgearing shenanigans): Moogle Ex (just the king), Garuda Ex (just the spiny plume), Ultima HM, Titan Ex, Ramuh Ex, Turn 6, CT second boss (I guess it counts?). I've heard of people swapping during T9 (like, in the middle of a claw so some hits hit one tank and some hit the other) when it was new, but I've only ever seen T9 solo tanked. To be honest, I'm a casual raider and haven't passed T9 yet, so I don't know about Final Coil.