The Chinese Rites controversy was a dispute among Roman Catholic missionaries over the religiosity of Confucianism and Chinese rituals during the 17th and 18th centuries. The debate centered over whether Chinese ritual practices of honoring family ancestors and other formal Confucian and Chinese imperial rites qualified as religious rites and were thus incompatible with Catholic belief.  The Jesuits argued that these Chinese rites were secular rituals that were compatible with Christianity, within certain limits, and should thus be tolerated. The Dominicans and Franciscans, however, disagreed and reported the issue to Rome.

 In Rome, the Jesuits tried to argue that these "Chinese Rites" were social (rather than religious) ceremonies, and that converts should be allowed to continue to participate. They maintained that Chinese folk religion and offerings to the Emperor and departed ancestors were civil in nature and therefore not incompatible with Catholicism, while their opponents argued that these kinds of worship were an expression of native religion and thus incompatible with Catholic beliefs.

Rome's Sacred Congregation for the Propagation of the Faith sided with the Dominicans in 1645 by condemning the Chinese rites based on their brief. However, the same congregation sided with the Jesuits in 1656, thereby lifting the ban.  It was one of the many disputes between the Jesuits and the Dominicans in China and elsewhere in Asia, including Japan and India.

The controversy embroiled leading European universities; the Qing dynasty's Kangxi Emperor and several popes (including Clement XI and Clement XIV) considered the case; the offices of the Holy See also intervened. Near the end of the 17th century, many Dominicans and Franciscans had shifted their positions in agreeing with the Jesuits' opinion, but Rome disagreed. Clement XI banned the rites in 1704. In 1742, Benedict XIV reaffirmed the ban and forbade debate.

Pope Clement XI's decree
Pope Clement XI condemned the Chinese rites and Confucian rituals, and outlawed any further discussion in 1704, with the anti-rites decree Cum Deus optimus of November 20, 1704. It forbade the use of "Tiān" and "Shàngdì", while approving Tiānzhǔ (‘Lord of Heaven’).

In 1705, the Pope sent a Papal Legate to the Kangxi Emperor, to communicate to him the interdiction of Chinese rites. The mission, led by Charles-Thomas Maillard De Tournon, communicated the prohibition of Chinese rites in January 1707, but as a result was banished to Macao.
Further, the Pope issued the 19 March 1715 Papal bull Ex illa die which officially condemned the Chinese rites.
 Pope Clement XI wishes to make the following facts permanently known to all the people in the world

    I. The West calls Deus [God] the creator of Heaven, Earth, and everything in the universe. Since the word Deus does not sound right in the Chinese language, the Westerners in China and Chinese converts to Catholicism have used the term "Heavenly Lord" 天主 (Tiānzhǔ) for many years. From now on such terms as "Heaven" 天 [Tiān] and 上帝 "Shàngdì" should not be used: Deus should be addressed as the Lord of Heaven, Earth, and everything in the universe. The tablet that bears the Chinese words "Reverence for Heaven" 崇拜天堂 should not be allowed to hang inside a Catholic church and should be immediately taken down if already there.

    II. The spring and autumn worship of Confucius, together with the worship of ancestors, is not allowed among Catholic converts. It is not allowed even though the converts appear in the ritual as bystanders, because to be a bystander in this ritual is as pagan as to participate in it actively.

    III. Chinese officials and successful candidates in the metropolitan, provincial, or prefectural examinations, if they have been converted to Roman Catholicism, are not allowed to worship in Confucian temples on the first and fifteenth days of each month. The same prohibition is applicable to all the Chinese Catholics who, as officials, have recently arrived at their posts or who, as students, have recently passed the metropolitan, provincial, or prefectural examinations.

    IV. No Chinese Catholics are allowed to worship ancestors in their familial temples.

    V. Whether at home, in the cemetery, or during the time of a funeral, a Chinese Catholic is not allowed to perform the ritual of ancestor worship. He is not allowed to do so even if he is in company with non-Christians. Such a ritual is heathen in nature regardless of the circumstances.
    Despite the above decisions, I have made it clear that other Chinese customs and traditions that can in no way be interpreted as heathen in nature should be allowed to continue among Chinese converts. The way the Chinese manage their households or govern their country should by no means be interfered with. As to exactly what customs should or should not be allowed to continue, the papal legate in China will make the necessary decisions. In the absence of the papal legate, the responsibility of making such decisions should rest with the head of the China mission and the Bishop of China. In short, customs and traditions that are not contradictory to Roman Catholicism will be allowed, while those that are clearly contradictory to it will not be tolerated under any circumstances.

In 1742 Benedict XIV reiterated in his papal bull Ex quo singulari Clement XI's decree. Benedict demanded that missionaries in China take an oath forbidding them to discuss the issue again.

In 1721, the Kangxi Emperor disagreed with Clement's decree and banned Christian missions in China. In the Decree of Kangxi, he stated,
   'Reading this proclamation, I have concluded that the Westerners are petty indeed. It is impossible to reason with them because they do not understand larger issues as we understand them in China. There is not a single Westerner versed in Chinese works, and their remarks are often incredible and ridiculous. To judge from this proclamation, their religion is no different from other small, bigoted sects of Buddhism or Taoism. I have never seen a document which contains so much nonsense. From now on, Westerners should not be allowed to preach in China, to avoid further trouble.'

In 1939, after two centuries, the Holy See re-assessed the issue. Pope Pius XII issued a decree on December 8, 1939, authorizing Chinese Catholics to observe the ancestral rites and participate in Confucius-honoring ceremonies.  The general principle of sometimes admitting native traditions even into the liturgy of the church, provided that such traditions harmonize with the true and authentic spirit of the liturgy, was proclaimed by the Second Vatican Council (1962–65).

Pope Pius XII's decision
The Rites controversy continued to hamper Church efforts to gain converts in China. In 1939, a few weeks after his election to the papacy, Pope Pius XII ordered the Congregation for the Evangelization of Peoples to relax certain aspects of Clement XI's and Benedict XIV's decrees. After the Apostolic Vicars had received guarantees from the Manchukuo Government that confirmed the mere "civil" characteristics of the so-called "Chinese rites", the Holy See released, on December 8, 1939, a new decree, known as Plane Compertum, stating that:
    Catholics are permitted to be present at ceremonies in honor of Confucius in Confucian temples or in schools;
    Erection of an image of Confucius or tablet with his name on is permitted in Catholic schools.
    Catholic magistrates and students are permitted to passively attend public ceremonies which have the appearance of superstition.
    It is licit and unobjectionable for head inclinations and other manifestations of civil observance before the deceased or their images.
    The oath on the Chinese rites, which was prescribed by Benedict XIV, is not fully in accord with recent regulations and is superfluous.
According to Smit this meant that Chinese customs were no longer considered superstitious, but were an honourable way of esteeming one's relatives and therefore permitted by Catholic Christians. Confucianism was also thus recognized as a philosophy and an integral part of Chinese culture rather than as a heathen religion in conflict with Catholicism. Shortly afterwards, in 1943, the Government of China established diplomatic relations with the Vatican. The Papal decree changed the ecclesiastical situation in China in an almost revolutionary way.  As the Church began to flourish, Pius XII established a local ecclesiastical hierarchy, and, in 1946, named Thomas Tien Ken-sin (Chinese: 田耕莘) SVD, then Apostolic Vicar of Qingdao, as the first Chinese national in the Sacred College of Cardinals and later that year appointed him to the Archdiocese of Beijing.


Early adaptation to local customs
The Jesuits of the Jesuit China missions made efforts to adopt Chinese customs.
Alessandro Valignano, Visitor of the Society of Jesus in Asia, was one of the first Jesuits to argue, in the case of Japan, for an adaptation of Christian customs to the societies of Asia, through his Résolutions and Cérémonial.

In China, Matteo Ricci reused the Cérémonial and adapted it to the Chinese context. At one point the Jesuits even started to wear the gown of Buddhist monks, before adopting the more prestigious silk gown of Chinese literati.  In particular, Matteo Ricci's Christian views on Confucianism and Chinese rituals, often called "the Directives of Matteo Ricci (zh)" (Chinese: 利瑪竇規矩), were followed by Jesuit missionaries in China and Japan.

In a decree signed on 23 March 1656, Pope Alexander VII accepted practices "favorable to Chinese customs", reinforcing 1615 decrees which accepted the usage of the Chinese language in liturgy, a notable exception to the contemporary Latin Catholic discipline which had generally forbidden the use of local languages.
In the 1659 instructions given by the Sacred Congregation for the Propagation of the Faith (known as the Propaganda Fide) to new missionaries to Asia, provisions were clearly made to the effect that adapting to local customs and respecting the habits of the countries to be evangelised was paramount.

    ‘Do not act with zeal, do not put forward any arguments to convince these peoples to change their rites, their customs or their usages, except if they are evidently contrary to the religion [i.e., Catholic Christianity] and morality. What would be more absurd than to bring France, Spain, Italy or any other European country to the Chinese? Do not bring to them our countries, but instead bring to them the Faith, a Faith that does not reject or hurt the rites, nor the usages of any people, provided that these are not distasteful, but that instead keeps and protects them.’    — Extract from the 1659 Instructions, given to Mgr François Pallu and Mgr Lambert de la Motte of the Paris Foreign Missions Society by the Sacred Congregation for the Propagation of the Faith.

In 1724 the Yongzheng Emperor proscribed the Heavenly Lord sect (天主教 Tianzhujiao, the name given Catholicism in China in that period).  Persecution steadily increased during the reign of the Qianlong Emperor.  While the Qianlong Emperor appreciated and admired the Jesuit Giuseppe Castiglione's artwork and western technologies, he also reinforced anti-Christian policies in 1737.

Source -  https://en.m.wikipedia.org/wiki/Chinese_Rites_controversy