The following quotes are from "That Christ is One," "Tome I Against Nestorius," and "Tome II Against Nestorius" which are written by a saint common to us all, Cyril of Alexandria.
This should show that the Oriental Church follows Miaphysitism. This also shows that the Eastern Church follows the Dyophysitism. Neither are teaching heresy, rather, they each developed a different way of understanding the mystery of the Incarnation.
Miaphysitism: Christ in one united nature (of two natures) in one person, his divinity united with his humanity in a real and perfect union without mingling, without confusion, without alteration, without division, without separation. The extreme is Eutychianism: Christ has one Nature of God and Man that is mixed and confused.
Dyophysitism: Christ in two natures united in one person, his divinity united with his humanity in a real and perfect union without mingling, without confusion, without alteration, without division, without separation. The extreme is Nestorianism: Christ has two separate Natures of God and Man that are one in dignity.
Cyril: Most mischievously therefore do they, severing into two the One and by Nature and Truly SON Incarnate and made Man, reject the Union and call it connection, which any one else too may have with God, being almost tied to Him by virtue and holiness, according to that which is by one of the Prophets rightly said to them who fall back into carelessness, Be ye gathered together and tied together, O undisciplined nation, ere ye become as a flower that passeth away: a disciple too may be connected with his teacher by means of love of learning, and ourselves, one with another, not in one way but in many. - St Cyril of Alexandria, That Christ is One
Hermias: Have they therefore been confused and both become one nature?
Cyril: But who will be thus distraught and unlearned as to suppose that either the Divine Nature of the Word has been turned into what it was not, or that the flesh went over by way of change into the Nature of the Word Himself (for it is impossible)? but we say that One is the SON and One His Nature even though He be conceived of as having assumed flesh with a rational soul. For His (as I said) hath the human nature been made, and He is conceived of by us none otherwise than thus, God alike and man. - St Cyril of Alexandria, That Christ is One
Hermias: There will then be not two natures, of God and of man?
Cyril: Godhead and manhood are one thing and another, according to the mode [of being] existing in each, yet in Christ have they come together, in unwonted wise and passing understanding, unto union, without confusion and turning. But wholly incomprehensible is the mode of the Union. - St Cyril of Alexandria, That Christ is One
Hermias: And how out of two things, Godhead and manhood, will One Christ be conceived of?
Cyril: In no other wise (I suppose) than that whereby the things brought together one to another unto a union indissoluble and above comprehension will be One. - St Cyril of Alexandria, That Christ is One
Hermias: But if we say that the Nature of the Son is One, even though He be conceived of as Incarnate, all need is there to confess that confusion and commixture take place the nature of man being lost as it were within Him. For what is the nature of man unto the excellency of Godhead?
Cyril: In highest degree, my friend, is he an idle talker who says that confusion and commixture have place, if one Nature of the Son Incarnate and made man, is confessed by us: for one will not be able to make proof thereof by needful and true deductions. But if they set their own. will as a law to us, they devised a counsel which they cannot establish, for we must give heed, not to them but to the God-inspired Scripture... - St Cyril of Alexandria, That Christ is One
Hermias: Know well that they will not choose so to think.
Cyril: Their speech will be caught setting forth to us most undoubtedly two sons and two christs.
Hermias: Not two: they say that the Son by Nature, the Word from forth God the Father is One; he that is assumed is a man by nature son of David, but is son of God by reason of his having been assumed by God the Word, and that by reason of God the Word dwelling in him hath ho come to this dignity and hath by grace the sonship. - St Cyril of Alexandria, That Christ is One
Cyril: Then wherever will they go as regards mind and understanding who thus think? or how do they say 'not a pair of sons,' when they are severing one from another man and God, if (according to them) the One has the sonship by Nature and truly, the other by grace and came to this dignity, God the Word indwelling him? Hath he then ought greater than we? for He indwelleth in us too. - St Cyril of Alexandria, That Christ is One
Nestorius: What then that is incongruous do WE say in advising to flee the word, and come to the common phrase significant of the two natures? then seems it to them that what is said is blasphemy. Either clearly acknowledge that the Godhead has been born of the blessed Mary, or if you flee this expression as blasphemy, why saying the same as I, dost thou feign thou sayest it not? - St Cyril of Alexandria, Tome I Against Nestorius
Cyril: And again, We recognise therefore the human nature of the Babe and His Godhead, we preserve the oneness of the Sonship in the nature of manhood and Godhead. - Cyril of Alexandria, Tome I Against Nestorius
Nestorius: But we must (for it has now come into my mind) learn that the Synod of Nicea too nowhere durst say that God was born of Mary; for it said, We believe in One God the Father Almighty and in One Lord Jesus Christ. Observe that having first put the word Christ, which is the indication of the two natures, they did not say, in one God the Word, but took the name that signifies both... - Cyril of Alexandria, Tome I Against Nestorius
Cyril: But since they say that they believe in One God the Father Almighty, Maker of all things both visible and invisible, and in One Lord Jesus Christ His Son, and none other (according to us) is Jesus Christ the Lord than the One and by Nature and truly Son... - Cyril of Alexandria, Tome I Against Nestorius
Cyril: That believing on Christ Jesus, we believe in the One and by Nature and truly Son, our faith mounting up unto the Father through Him, will be clear, in that He Himself hath cried aloud to the whole world, He that believeth on Me believeth not on Me but on Him that sent Me, and he that seeth Me seeth Him that sent Me, and again, Believe on God, and believe on Me. - Cyril of Alexandria, Tome I Against Nestorius
Cyril: But WE, my friend, who call her mother of God, have never at all deified any one of those that are numbered among creatures, but are accustomed to know as God the One and by Nature and truly so: and we know that the blessed Virgin was woman as we. But thyself wilt be caught, and that at no long interval, representing to us Emmanuel as a God-bearing man, and putting upon another the condemnation due to your own essays. - Cyril of Alexandria, Tome I Against Nestorius
Cyril: But the inventor of the most recent impiety, albeit making feint of saying One Christ, ever divides the Natures and sets Each by itself, saying that they did not truly come together... - Cyril of Alexandria, Tome II Against Nestorius
Cyril: For the Incarnate Nature of the Word Himself is after the Union now conceived of as One, just as will reasonably be conceived in regard to ourselves too, for man is really One, compounded of unlike things, soul I mean and body. - Cyril of Alexandria, Tome II Against Nestorius
Cyril: And I will for profit's sake add this too: other than the Word out of God is the flesh, in regard to its proper nature, other again Essentially the Nature of the Word Itself. But even though the things named be conceived of as diverse and sundered in diverseness of nature, yet is Christ conceived of as One out of both, the Godhead and manhood having come together one to another in true union. - Cyril of Alexandria, Tome II Against Nestorius
Nestorius: But to speak clearly and more intelligibly to all, it is the aim of the party of Arius and Eunomius and Apolinarius and of all who are of like brotherhood, to bring in Theotokus, as though, a mingling having taken place and the two natures not divided... - Cyril of Alexandria, Tome II Against Nestorius
Cyril: Is not he other than the Word, One and by Nature and forth of God the Father, who in his proper person has been verily parted from union with Him? and how is this not clear to all? - Cyril of Alexandria, Tome II Against Nestorius
Nestorius: When therefore the Divine Scripture is about to speak of either the birth of Christ which was forth of the blessed Virgin, or His Death, it is never seen to put God, but either Christ or Son or Lord, seeing that these three are significant of the two natures, one while of this, other while of that, other while of this and that. - Cyril of Alexandria, Tome II Against Nestorius
Nestorius: But since the Son of God is two-fold in His Natures, she bare indeed the Son of God, but bare the manhood which is son by reason of the connected Son. - Cyril of Alexandria, Tome II Against Nestorius
Cyril: Thus are WE minded to think and are accustomed to walk aright, going on the royal and unperverted road: but he saying that such names are indicative of the two natures, allots to either with authority what seems good to him and is ashamed of the lowliness of speech belonging to the economy with flesh... - Cyril of Alexandria, Tome II Against Nestorius
Nestorius: For God (he says) sent forth His Son made of a woman, if made under the law. Here he points out the two natures, he says what took place as to the human nature, for demand of the wrangler, Who was made under the law? was it God the Word? - Cyril of Alexandria, Tome II Against Nestorius
Cyril: Therefore (as saith the most wise Paul) the powers above are bidden to worship the First-begotten when introduced by the Father into the world, and on learning the mystery regarding Him, with ceaseless praises do they extol the One and by Nature and truly Son. - Cyril of Alexandria, Tome II Against Nestorius
Nestorius: Say of Him Who assumed that He is God, add of that which is assumed that it is the servant's form, bring in next the dignity of connection, that of the two the sway is common, that of the two the dignity is the same; while the natures remain, acknowledge the union of rank. - Cyril of Alexandria, Tome II Against Nestorius
Nestorius: Therefore I would have you hold fast with all assurance: there is no severance of the connection of the dignity of the sonship, there is no severance of his being Christ, of the Godhead and Manhood there is a severance; Christ is indivisible, in that He is Christ, for we have not two christs nor two sons, for there is not with us a first and a second, nor yet other and other, nor again another son and another again; but the One is Himself twofold, not in rank but in nature. - Cyril of Alexandria, Tome II Against Nestorius
Cyril: One both before flesh and with flesh: for thus will He be in respect of rank (as THOU sayest) and also of sway, inseverable, yea rather the Same. Then how dost thou say that the One and Inseverable is twofold, and that not in regard of rank but of nature? for not because the Word out of God the Father having taken flesh, proceeded forth man as we, will He for this reason be called also twofold, for One and that not without flesh is He Who is in His proper Nature external to flesh and body. - Cyril of Alexandria, Tome II Against Nestorius
Nestorius: But since he has been connected with Him which is in the beginning Son, Him who was connected with him, he may not admit of severance in respect of the dignity of sonship, in respect I say of the dignity of sonship, not in respect of the natures. Wherefore God the Word is called Christ also, since He has His connection with Christ perpetual. - Cyril of Alexandria, Tome II Against Nestorius
Cyril: He that durst say that the good glory of the Only-Begotten has been ascribed to the power of Christ, and that plucked asunder the bond of Oneness, gathers again into union and again dissolves it and parts the natures one from other. - Cyril of Alexandria, Tome II Against Nestorius
Cyril: If thou sayest that the Word of God was made man, this will suffice to shew that He Who is above all the creation was made as we. He took the servant's form although He possessed freedom as God; for He was in equality with the Father, Who possesseth dominion over all. Cease to sever the natures after the union: for that one thing and another is the Divine Nature and the nature of man it will be fit to know, and needful I deem to those who are sound in mind (for they are parted one from another by incomparable differences), but in regard to Christ the Saviour of us all, do thou having brought them together into union true and of Person, reject severance, for thus wilt thou confess one Christ and Son and Lord. - Cyril of Alexandria, Tome II Against Nestorius
Cyril: Yet if you say One Son and One Person, the Incarnate Person of the Word, He will not be an instrument of Deity, but rather will use as an instrument His own Body, just as a man's soul too does. Therefore confess One, not dividing the natures... - Cyril of Alexandria, Tome II Against Nestorius
Cyril: Thou again completely severest and dealest subtilly with the truth, parting the natures, uniting (as you say) the worship. But if you part the natures, along with them will diverge the natural properties too of either, the count of their difference will speed apart: hence two are they confessedly. - Cyril of Alexandria, Tome II Against Nestorius
Nestorius: Keep we therefore unconfused the connection of the natures, confess we One God, reverence we the man who is co-worshipped by a Divine connection along with the Almighty God. - Cyril of Alexandria, Tome II Against Nestorius