I know there’s local (name removed by moderator)ut, which is great, but my question is more around the formal appointment itself which ultimately comes from Rome.
Its actually the Apostolic Nuncio in each country who generates a shortlist of three names which he forwards to the Congregation for Bishops in Rome.
There’s a bit of history behind all that.
Originally (in very brief - so anyone can just insert in anything you feel significant that I have left out), all bishops are elected locally or sent from a nearby sponsoring church. After a while the Roman bishopic started to exert its judicial authority in the West and began to confirm the elected bishops in the West. Early church councils also provided for the metropolitans to have a role in confirming the elected bishops. Patriarchs from the East are not confirmed by the Pope but all five patriarchs enters the other patriarchs into a list on their election to signify his communion with them.
After Christianity became the official religion in the Empire, the Emperor began to have a role in appointment of bishops on the grounds that the bishops have a fundamental role in maintenance of peace and order in the realm. This continued after the fall of the Roman Empire with the influence/power being exerted by the local potentate. The degree of royal influence in the naming of bishops differs with the local situation/politics/royal power and the charisma of the individual Pope. In the East for instance, where papal writ did not run so far and the Emperor remain supreme, the Byzantine emperor named the bishops especially the Patriarchs of Constantinople.
By the middle ages many of the bishops in the Western church are appointed by the local king, or they at least hold a veto. Vestiges of these still remain in episcopal appointments in the Church of England. Today, English bishops are selected by the Queen from a list of two names proposed by the Prime Minister. Which was why Tony Blair only converted to Catholicism after his resignation as while it is politically feasible for a non-Anglican Protestant to name Anglican bishops, it is not the done thing for a Catholic PM to do so. (So much for an enlightened non-discriminatory age) You will find that much of the structures in the Church of England remains frozen at the time of the break while the Roman Catholic church has moved on.
An opportunity to break the power of royal patronage over episcopal appointments presented itself with he age of exploration. The Portuguese and Spanish crowns continued to appoint bishops in their colonial lands but the Pope started to appoint directly bishops in lands evangelised by Jesuit/Dominican/Franciscan missionaries on the grounds that there is no Catholic monarch there. This was normally done by designating the head missionary as a Vicar Apostolic and setting up a Vicarate Apostolic which is a bishopric in all by name but without most of the diocesan faculties normally found in a diocese. This avoids the sensitivities of the Portuguese and Spanish crowns. Vicarates Apostolic continue to be the precursor of a diocesan hierarchy today, especially where there are local political sensitivities in having Catholic bishops (eg., in some Muslim countries).
This is where the separate roles by the Congregation of Bishops and the Congregation of Propagation of Fatih have in nominating bishops until today. The Congregation of Bishops gradually took over the role from the European monarchs. I believe the Astro-Hungarian emperor was the last to hold a theoretical veto until the 19th century. Propaganda Fidei continued its role from the early days and now nominates most of the Catholic bishops in the world. There is a third Vatican Curial congregation (Congregations are equivalents of US Departments in the Vatican) but that is for the Eastern Catholic churches.
The procedure is very much like you mentioned - just to add that the three names are provided annually and always in secret. The candidate will have no idea that the nuncio/apostolic delegate named him. The nuncio/apostolic delegate are considered the best trained diplomatic corps in the world and do a great job keeping tabs on the church and local political situation (episcopal appointment often have to consider the local political situation with the populace and the government in addition to pastoral considerations). The Pope is considered to have one of the most extensive intelligence network in the world and certainly the most efficient.
There are variations to this model. There are a number of German and Swiss bishoprics (I think about half of them) plus Salzburg in Austria (of Sound of Music fame) where the local cathedral chapters either provides the three names to the Pope or elects the bishop from the three names provided by the Pope.
I think that’s all I know about the subject


. Have a blessed Christmas.