All sacramental grace flows from its source - the passion, death and resurrection of Our Lord - through His Church, which is the mystical extension of the Incarnation - the body of the Lord on earth. Jesus instituted His Church to continue the work of redemption that He began. With this in mind, we must then acknowledge that any valid eucharist must in some way come from and through the Church. Our Lord established the sacraments to work in a certain way. Think of it like the laws of nature - God ordained that when we drop something, gravity kicks in and the object falls to the ground...this always happens no matter who drops the object.
The sacraments, in a sense, operate under spiritual laws ordained by God when He instituted the Church. This ensures that we can always rest assured that we will receive graces and encounter the Risen Lord when we receive the sacraments, regardless of the interior disposition of the minister. Eastern Orthodox priests, Oriental Orthodox priests, Assyrian Church of the East priests, and the priests of a number of other smaller bodies not in full communion with the Catholic Church, have validly received the sacrament of holy orders from a bishop who can trace his apostolic succession, through the laying on of hands, back to a bishop who was, at one point, in full communion with the Catholic Church and a true successor of the apostles. God respects our free will - the authority that Christ gives His bishops through the sacrament of holy orders can not be taken away. While ordaining a priest while not in full communion with Rome and the Catholic Church may not be according to God's will and may be sinful (if done with full knowledge and consent of the will), the bishop's imperfect intentions do not invalidate the sacrament - he has the authority. Holy Orders has configured his soul in such a way that he is a bishop just as much as you and I are human beings. Just as you and I can always walk or talk or eat, regardless of whether we use these abilities according to God's will, so can a bishop exercise his apostolic authority to ordain priests and confect the eucharist.
So in short, the eucharist offered in the Orthodox and other non-Catholic churches with valid apostolic succession, still ultimately flows from the Catholic Church, but the individual ministers are in an imperfect union with her. The Catechism says:
838 "The Church knows that she is joined in many ways to the baptized who are honored by the name of Christian, but do not profess the Catholic faith in its entirety or have not preserved unity or communion under the successor of Peter." Those "who believe in Christ and have been properly baptized are put in a certain, although imperfect, communion with the Catholic Church." With the Orthodox Churches, this communion is so profound "that it lacks little to attain the fullness that would permit a common celebration of the Lord's Eucharist."
Think of it this way...by virtue of my baptism, I share in the common baptismal priesthood of all believers. This means that I have the authority to pray in Christ. When I fall into mortal sin, by an act of my will, I am no longer in perfect communion with Christ and His Church...but I still remain a part of the Church by virtue of the indelible mark baptism left on my soul, and while my prayers may not carry the same weight, I still am able to pray.