There are different levels of AI diplomatic threat. Threats can be weak implied threats of do this "or else", (basically a poker style bluff), or strong demands of do this or we will attack you.
How the AI react to your response will depend on your existing diplomatic relations with them and the strength or weakness of your military.
The ultimate appeasement gesture is to give up a human controlled city, so as to buy peace or ward off an invasion. The good / bad consequences of such appeasement is situationally dependent. If the city is a border city you have captured from an enemy, of little or no strategic values, which you can not defend, then I am happy to use appeasement to stop a war. If the city is a core inner city that I built, or a captured city that has high strategic value, then appeasement is bad.
Generally speaking you can use appeasement to your advantage, but you have to work the diplomatic screen to get the maximum benefit. For example, I remember one game on Emperor, where I shared a continent with 3 civs, but had a very poor starting position. I therefore attacked one civ to gain room and resources and I eventually reduced them to circa 3 cities, but that left me massively overstretched, and very vulnerable to the other 2 civs, with whom I had poor diplomatic relations. Each of them had made demands of me and I gave in to all the demands they made. The civ closest to my core cities was the stronger, and so I gave away / liberated two captured border cities to the weaker civ. This increased their weak power. I adopted their religion and civics, and gave away tech and trade goods to massively improve my diplomatic relations with them and increase their power. That civ then attacked the other one, rather than me, and as both these civs were now equally balanced, neither of them was able to obtain swift gains. This gave me the time and the breathing space to consolidate my existing position, and during that time I gave into yet more outrageous demands. I eventually joined in the war to win a few more cities from the closest civ, and I then back stabbed the civ I had appeased earlier with cities, techs and gifts.
I ended the game in control of the continent, and won a Space victory, but I had used appeasement as part of an overall strategy, rather than as a knee jerk reaction to a one off threat made out of the blue.
You can therefore successfully use appeasement on a huge scale in the right situation, but only when it is an integral part of a greater overall strategy.
Regards - Mr P
