When and why did the Catholic Church stop the Crusades?

(TL;DR): Well, the Catholic Church didn't call a stop to the Crusades. It is the crusaders themselves who gave up after the failure of the siege of Tunis.

As user13992 said, Crusades "stopping," we are not talking of an ongoing Crusade being called off, but of a new Crusade not being convoked, but, furthermore, the Catholic Church didn't call to stop the crusades neither.

The first crusade took place from 1096 to 1099 following, among other things, the refusal in 1078 of the Seljuk Turks to give free passage to christian pilgrims to Jerusalem.

On November 27, 1095, Pope Urban II called to arms all Christendom. He evoked the "misfortunes of Eastern Christians". He called on christians in the West to stop waging war and to unite to fight the "pagans" (need reference) "and deliver the brothers of the East.

This crusade ends with the capture of Jerusalem and the creation of the Christian kingdom of Jerusalem. But in the West, the crusade continues to be preached. Pope Paschal II pronounces the excommunication against those who have not fulfilled their vows, sending deserters back to Jerusalem, such as Stephen de Blois and Hugues de Vermandois. Which translates into taking Ascalon.

The initiative of the second crusade returns to King Louis VII. He wanted to go on pilgrimage to Jerusalem to atone for his faults: a crime whose memory tormented him: the burning of the church of Vitry-en-Perthois in which more than a thousand people died. He obtained from the pope the new promulgation of a crusade bubble, hitherto without effect. The preaching goes back to Bernard from Clairvaux to Vézelay on March 31, 1146 then to Speyer. In Germany, the popular preaching of a former Cistercian monk provokes a new outburst of violence against the Jews. It ended in 1149 with a total failure for the Crusaders, who returned to Europe without having won a military victory in the East. Conrad III and Louis VII then return to the West without any military victory. I have no source on the position of the Catholic Church who would ask to stop the crusade.

The third crusade, which began in 1189 and ended in 1192, is a series of expeditions led by Frederick Barbarossa, German Emperor, Philip Augustus, King of France, and Richard the Lionheart, King of England, for the purpose to take back Jerusalem and the Holy Land to Saladin. The third crusade is launched by the call of Pope Gregory VIII, who decides to reconquer the lost territories and issues on October 29, 1187, the bull Audita tremendi, calling for a third crusade. It ended with the free access to Jerusalem without taxes, as well as the free movement of merchants of the two denominations inside the city is concluded on September 2, 1192. Richard leaves the Holy Land on October 9 to return to the West. I have no source on the position of the Catholic Church who would ask to stop the crusade.

The fourth crusade was called by Pope Innocent III in 1202. From the beginning of his pontificate, he wanted to launch a new crusade to the holy places of purely pontifical inspiration. He forges the idea of ​​"political crusades" that will be taken over by his successors. He raises the first of the taxes to finance the crusades and expresses first the right to "the exhibition of prey", that is to say the right for the pope to allow Catholics to seize land from those who would not repress heresy. The crusaders gather in Venice but can not pay their trips to the Venetian shipowners, are diverted by them to Zara on the Dalmatian coast they besiege and take on behalf of Venice. The pope excommunicates the Crusaders and Venice but quickly raises excommunication for the Crusaders. The deviation from the very idea of ​​crusade and the pillaging of Christian Constantinople transforms military orders into financial and, consequently, political powers. The Church then no longer has the monopoly of the call to the crusade. And I have not found any writings calling for an end but these excommunication documents.

The sixth crusade is made because of the fifth, which was first successful with the capture of Damietta and then with the acceptance, by the Sultan of Egypt, to exchange Damietta against Jerusalem. But the intransigence of Legate Pelagius had caused the crusade to fail and had made him lose all these advantages. John of Brienne, King of Jerusalem, goes to Rome to discuss with Pope Honorius III conditions for a new crusade. He married his daughter to Frederick II while the Pope ordered Frederick II to go on a crusade, under pain of excommunication. Frederick II gets through the negotiation Jerusalem. While Frederick II went to the East to fulfill his promise to take up the cross, the pope launched against him an army financed by a tax on the income of the clergy and leftovers. In 1237, a new crusade was launched by Pope Gregory IX and got more land, but most of the armies are those of earls and barons, the Church is then dumb.

The church did not call the seventh crusade nor the eighth, led by St. Louis, and did not call for stopping it either.

Finally the end of the Crusades comes to the crusaders who, following the failure of the siege of Tunis agree to return to France to prepare for a new crusade that will never happen according to Jean Richard in his biography of Saint Louis.

As a conclusion, if the Church had sometimes called to take up the crusade and especially in it's origin; it didn't make a call to stop them. It has been dispossessed of the power to call the crusade as it weakens.

My opening question would then be to what extent can it be said that there was never a ceasefire between the Catholic Church and those she called to fight in the same way as there wasn't one between North Korea and South Korea after the end of the fighting in 1953?

Most of the credit for these research are given to French pages of Wikipedia, Croisades, which I picked and translated here and there.