the history of the Church, namely the bishops, having authority over marriage of Christians begins with the apostles and early fathers. the NT in Acts and in Pauls letters gives apostolic instruction on marriage, including two rules now called the Pauline and Petrine privilege relating to the dissolution of marriages between Christians and non-believers for instance. I have not time now to look up references but will this evening when I get home from work. I believe a search on the home page on This Rock will yield quotes from the Early Fathers on marriage.
there is no such thing as annulment. The canon law tribunal of the diocese investigates a marriage, a legal process like any legal investigation with witnesses, lawyers and judges, and issues a decree of nullity if the marriages is found, under canon law, to be invalid. The history of what we commonly call annulment therefore as a legal process begins with the history of the development of formal codified canon law. For this see a good general church history. This development began when the bishops of Rome, the Popes, gradually took over some of the civil functions of the Empire as it crumbled in the West and the legions deserted and left Romans to the barbarians.
The various revisions of canon law, including the general one in 1917 and the latest in 1983, plus some commentaries on the law published recently (last 100 years is recent in Church history) have led to the way the process is administered now.
the development of the canon law cannot of course be divorced (npi) from the development of doctrine and theology relating to the sacraments, especially marriage, and of the Eucharist from which it derives its grace and its analogy to the relationship between Christ and His Church. the underlying ground for the insistence of the Catholic Church that a valid marriage may not be dissolved by any earthly action or power rests on Christ’s declaration of the same, except as he specifies in cases of illegality, which the Church takes to mean, in cases where the marriage is not valid from its inception. IMO Christ was being prophetic here in anticipating development of law related to the sacraments.
The teaching of the Orthodox is similar and comes from the same roots, and the differences, as far as I can see and I am no expert, lie in the definition of “illegality”. I do not believe the Orthodox bishop would recognize civil divorce alone as dissolving a marriage, but that there is some church law procedure involved as well.
as far as Protestants, most mainline denominations adhered to the gospel principal until the Lambeth Conference in the 1930s (convocation of Anglican leaders) “legalized” both divorce and artificial contraception. They will have to answer to God for those actions, beyond the scope of current discussion.