Protestants, contra their Roman Catholic and Greek Orthodox counterparts, disassociate themselves, for the most part, from the concept of Apostolic Succession. This movement away from this concept was re-initiated most succinctly, and with shaping force, by the “Reformers.” Anglican, Paul Avis provides a good synopsis of how the Reformers distanced themselves from this defining ecclesial feature of Roman Catholics:
The concept of the Church which was fundamental to the thought of the Reformers (including of course the Anglicans)—namely, that only the gospel was of the esse—had profound implications for the doctrine of succession and with it the key concept of catholicity, one of the four creedal attributes of the church. Here a radical reinterpretation was effected. . . . By making the gospel alone the power at work in the Church through the Holy Spirit, the Reformers did away with the necessity of a doctrine of apostolic succession, replacing it with the notion of a succession of truth. Correspondingly, the gospel of truth was held to be sufficient to secure the catholicity of the Church. The Reformers believed with all of Christendom that the Church was one, holy, catholic and apostolic, but this was understood in a radically new sense in which the gospel itself became the decisive and dominant criterion. (Paul Avis, “The Church in the Theology of the Reformers,” 127-28 )
Of course the “Gospel of Truth” was assumed to be clearly comprehended by interacting with the “Apostolic Witness,” deposited for the Church, in the writings of the New Testament (and Old Testament). It was these writings that the Reformers believed “clearly” (the a priori commitment of the “Reformers” was in the Perspicuity of the Scriptures) bore witness to The Gospel, The Person, The Head of the Church—Jesus Christ.
How does this differ from the Roman conception of “Apostolic Succession?” It differs in the sense that it sees the Gospel, itself, better, “Himself,” as The esse, constitutively, of the concrete shaping of the Church. In other words, the very trinitarian life of God, as disclosed in the person of Jesus Christ becomes the cornerstone for “who” the Church “catholic” (universal) is. The “Church” is the Church by an immediate union with Christ, instantiated by the wooing work of the Holy Spirit, through the instrumentality of the Holy Scriptures, made known through the proclamation of the Church. So in this sense, the concrete shape of the Church is not isolated to an “office of succession,” intrinsically tied to the Roman expression of the Church; instead the concrete shape of the Church is essentially rooted within the person of Jesus Christ as “actualised” in the incarnation, and realized behind the Church who proclaims His life shaping message. In this accounting, then, the Church is not necessarily tied to a particular “church,” but the Church becomes the Church, apart from any “organizational structure,” through her reception of the Gospel, as determined by the purposes of God of God.