Did some Christian sects ever hold the opinion that ritual slaughter was obligatory, before the 10th century? Like this question, I am curious about whether Christian doctrines used to hold the opinion that a ritual slaughter (with it rules and conditions) had to be done when slaughtering animals, historically speaking.

In Islam (and Judaism) there are prohibitions about eating dead animals, which haven't been slaughtered in a certain way. Muslims may for instance eat meat if the animal was slaughtered by a Christian if it was slaughtered in accordance with those rules. 

If Muslims were allowed to eat meat slaughtered by Christians, that should imply that the Christians did have a method of ritual slaughtering. (Because according to the Islamic principles, a Muslim may only eat meat which has been slaughtered right according to some special conditions or rules, therefore if a Muslim may eat meat slaughtered by Christians it should imply that the Christians did ritual slaughtering, if not their meat wouldn't be allowed for Muslims. But it is.)

Are there any known Christian school of thoughts or historical evidences that Christians used to slaughter in a specific way, and that they prohibited certain meat (not slaughtered right or for instance swine)?

In order to make the question less broad, I am looking for the early period of Christianity, up to the 10th century (but around the 6th century is preferable). The place could of course also be factor so I am adding these into the equation: Arabia, Egypt, Jerusalem, Iraq.
I do not know if this response will actually answer your question, but I will give it a possible conclusion to it.

In a sense animal sacrifice and ritual slaughter are not always clearly seen or defined, yet it would seem inappropriate to have a animal sacrifice without a set ritual for slaughtering such animals.


  Rituals of animal sacrifice have been far more common in Christianity than is generally recognized. When missionaries arrived in Britain around the year 600, sacrifice was so deeply engrained in culture that the Pope allowed it to continue provided that the animals were offered to the Christian God and no other. In more recent centuries, these rituals have been mainly confined to Eastern Orthodox Churches. An ancient sacrificial liturgy known as the matal, meaning ‘something tender’, has persisted In the Armenian Church to the present day.
  
  A key requirement of animals offered for sacrifice is that they must be unblemished, ensuring that they have been well-fed, comfortably accommodated and free from disease. In the Armenian matal, at least three factors minimize the suffering caused to the animal during slaughter: rules must be observed, attentiveness shown by the slaughterer, and reverence displayed at all times. In some accounts of sacrifice, the animal is presented as so contented that it co-operates in its own sacrifice. Paulinus (354-431) describes such an instance at the shrine of Felix of Nola in southern Italy, in which a heifer designated for sacrifice guided her owners to the shrine to present herself with them. - Christian Attitudes to Animals


There is the Armenian Matagh tradition that persists to this day. It should be noted that according to tradition, Armenia became a Christian nation in the Year of Our Lord 301.


  In Armenian Christian tradition, matagh (Armenian: մատաղ mataġ) is a lamb or a rooster slated for sacrifice to God, a ritual which has continued from the pagan past. In many regions of Armenia today, this pagan-Christian synthesis is very much alive in the regular slaughter of chosen animals in front of churches. Matagh is done often to ask God for either forgiveness, health, or to give them something in return. People generally gather at the house where the Matagh was done, where they pray and eat the meat. Tradition holds that the meat must be eaten before sundown.


It is not impossible that the Ethiopian tradition of slaughtering animals while saying the trinitarian formula predates the 10th century.


  In the 7th century the conquests of the Muslim Arabs cut off the Ethiopian church from contact with most of its Christian neighbours. The church absorbed various syncretic beliefs in the following centuries, but contact with the outside Christian world was maintained through the Ethiopian monastery in Jerusalem. - Ethiopian Orthodox Tewahedo Church


Wikipedia has this to say about Christian dietary laws:


  Slaughtering animals for food is often done with the trinitarian formula. Although the Armenian Apostolic Church, among other Orthodox Christians, have rituals that "display obvious links with shechitah, Jewish kosher slaughter." In addition, meat consumed by Christians should not retain any blood, a practice that both Jewish and Islamic methods of slaughter also prescribe, and one that is done by most slaughterhouses throughout Christendom. - Christian dietary laws


Today this tradition can still be seen in Ethiopia, the only question is as to when this tradition was first implicated.


  In Addis Ababa, there is one Christian slaughterhouse and one Muslim one, each of which supplies all respective butchers and restaurants. At the Christian slaughterhouse, an Orthodox priest will bless all the animals with a Trinitarian blessing, a pattern that is repeated in other large towns and cities. In the countryside, this may be left to the senior male householders who pray a Trinitarian blessing over the bull, goat, lamb or chicken before its throat is cut. - Food &amp; Fasting