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Historically, the fact that the birth of the church is associated with the "arrival" of the Holy Spirit in Acts 2 at Pentecost, is enough for most people to more closely associate the Church with the Spirit. Likewise, the rest of Acts ascribes the growth of the church moreso to the Spirit than to Jesus. Finally, Jesus mentions the work of the Paraclete and Comforter as the one He would send to us, and as such, thus adheres more closely to the Spirit than to the Son.
As such, Ecclesiology is thus more closely related to Pneumatology than to Christology.
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It depends. Ecclesiology in Eastern Orthodox Church is predominantly based from their Pneumatology that the Holy Spirit proceeds eternally from the Father alone. In this sense, the visible Church is connected invisibly to the Holy Spirit who proceeded not from the Son. This is why decentralization is the dominant view in Eastern Orthodoxy. In the other hand Oriental Orthodox is influenced by their miaphysite Christology that the Church is united as a compound unity. Like the unity between soul and flesh in psychosomatic unity. Soul and flesh are no longer two after union but united as one compound unity. This is why each Oriental churches is independent. EO and OO share a similar view of ecclesiology. But because EO draws their ecclesiology from pneumatology there is theological development such as Palamism and Toll Houses. While in OO because after union there is only one compound nature the Church is static, this is why there is no serious theological controversy in OO history because their theology stay the same after Ephesus.
In Catholic Church, both Latin and Eastern rites, we see our ecclesiology not from Pneumatology or Christology particularly but from Triadology. How the Father, the Son, the Holy Spirit relate to one another in hierarchical harmony. This is where we get the notion of Ecclesiastical Hierarchy it's seen from Trinitarian relationship. From Trinitarian taxis (eg. ordering of three persons) we have universal (eg. Pope), regional (eg. College of Bishops), and local (eg. local bishop) primacy. Each share the same ecclesial essence. Because the Holy Spirit proceeds from the Father through the Son, each local church is independent while maintaining unity through communion with the Church of Rome, which is one out of 24 churches within Catholic communion who presides in love.