After some research, it appears this is how it works with monsignors:

"A Catholic priest is given the title of Monsignor by the Pope in recognition of outstanding service to the Catholic Church. A monsignor is still a priest but also has this great honor to his name.

The process begins once a diocese bishop nominates a priest for this high honor. The bishop submits the priest's name, biography and works to the Holy See. At that point, the Pope reviews the information and determines whether or not to make the priest a monsignor. The Vatican will issue a diploma designating the new title and rank.

There are three grades of monsignor. The Protonotary Apostolic is given to a priest who serves seven positions in the Roman Curia. A Protonotary Apostolic wears a black cassock with red buttons and a fuchsia sash and cape. A grade 2 monsignor is Prelate of Honor to His Holiness. It can be given to priests outside the papal court. The Prelate of Honor to His Holiness wears a bishop's choir cassock and a fuchsia sash. The Chaplain to His Holiness is grade 3 and is given to a priest inside or outside the Roman Curia. They wear a black cassock with a fuchsia sash."

Is it proper to refer to a priest who is a monsignor who becomes bishop as a monsignor?

I have heard it done before I think.

From Catholic Encyclopedia:

(From mon, "my" and seigneur, ("elder" or "lord," like Latin senior)

A French honorific appellation, etymologically corresponding to the English "my lord," and the Italian monsignore. It is, after all, nothing but the French monsieur; but, while the latter has become current as applied to every man who is in good society, Monseigneur has retained its honorific force. In ecclesiastical usage it is reserved for bishops and archbishops, and is chiefly employed when speaking or writing to them. It is used before the name (thus abridged: Msgr. Dupanloup). Formerly it was not prefixed to the title of dignity, but it is now, as "Mgr l'évêque N. . . ." The term Monseigneur is also used as the equivalent of the Italian Monsignore, and as the latter title is given to Roman prelates, some confusion results; in Italy, however, no inconvenience arises from this usage as in that country bishops have the title of Eccellenza, i.e., Excellency. In France, only the Archbishop of Reims, as legatus natus, has the title of Excellency (See MONSIGNOR).
 

(Dominus meus; monseigneur, My Lord).

As early as the fourteenth century it was the custom to address persons high in rank or power with the title Monseigneur or Monsignore. In the intercourse of seculars, either of equals or of superiors with inferiors, there was no fixed rule. Until the seventeenth century French nobles demanded from their subjects and dependents the title of Monseigneur. In international intercourse two titles gradually won general recognition, "Monsieur" as the title of the eldest brother of the King of France (if not heir presumptive) and "Monseigneur" for the Dauphin, or eldest son of the French king, who was also the crown prince, or for whatever male member of the family was recognized as heir presumptive to the throne. Actually all Bourbon pretenders assume this title as a matter of course, e.g. the late Don Carlos Duke of Madrid, his son Don Jaime, the Count of Caserta, the Duke of Orléans, etc. Moreover, the custom often obtains, especially in Spain, France, and Italy, of extending by courtesy the title Monseigneur to the adult members of the Bourbons and closely allied families usually addressed as "Your Royal Highness". In official usage, however, this would scarcely be permissible. At present the title is no longer borne by other persons of civil rank, and, so far as the author of this article is aware, no one else lays claim to it. Among ecclesiastics the title Monsignore implies simply a distinction bestowed by the highest ecclesiastical authority, either in conjunction with an office or merely titular. In any case it bears with it a certain prescribed dress. To counteract a widely spread misconception we may state here that the pope does not bestow the title Monsignore, but a distinction of some sort to which this title is attached. Accordingly it is quite incorrect to say that any one has been appointed a Monsignor by the pope. If we may be permitted to use a comparison, Monsignor in the spiritual order corresponds to the word officer in the military. The highest general and the youngest lieutenant are equally officers, and the most venerable patriarch bears the title Monsignor as well as the simplest honorary chaplain. Thus among prelates, both higher and lower, it is no badge of distinction except as it denotes in a very general way an elevation above the ranks of the clergy. Those only bear the title of Monsignor, who are familiares summi pontificis, those who, by virtue of some distinction bestowed upon them, belong as it were to the family and the retinue of the Holy Father. These familiares are entitled to be present in the cappella pontificia (when the pope celebrates solemn Mass), and to participate in all public celebrations purely religious or ecclesiastical in character, at which the pope, the cardinals, and the papal retinue assist. it is assumed that they will appear in the robes corresponding to their respective offices.

Up to 1630, when Urban VIII reserved the title Eminence (Eminentissimus) for the exclusive use of cardinals, the latter bore the title of Monsignor in common with the other prelates of high rank, and in France it is still customary to address a cardinal as Monseigneur. In all other languages this usage has completely disappeared, so that, practically speaking, cardinals are no longer counted among the Monsignori. All other prelates, from patriarchs down, who have received a papal distinction or are archbishops, bishops, or mitred abbots (among the secular clergy only), have a right to this title. The fact that it lapsed in usage in many countries, so far as these are concerned, does not affect the question. Instead of addressing patriarchs as "Vostra Beatitudine", archbishops as "Your Grace", bishops as "My Lord", abbots as "Gracious Lord" one may without any breach of etiquette salute all equally as Monsignor. Following is a list of official and honorary prelates exclusive of those already mentioned:
•the college of the seven official prothonotaries Apostolic de numero participantium (of the number of participants);
•the supernumerary prothonotaries (supra numerum), including, (a) the prelate canons of the three patriarchal basilicas of Rome, (b) the prelate canons of certain cathedral churches, while in office;
•prothonotaries Apostolic ad instar participantium (after the manner of participants), including, (a) prelate canons of certain cathedral churches, as above, (b) prothonotaries appointed ad personam (individually);
•the College of Auditors of the prelates;
•the college of official clerics of the Apostolic Camera;
•all other prelates not members of any of the above named colleges, the numerous domestic prelates scattered throughout the world. All the above-mentioned prelates are entitled to wear the mantelletta and rochet;
•the private chamberlains constituting the official college of pontifical masters of ceremonies;
•the official private chamberlains known as participantes;
•the super-numerary private chamberlains (camerieri segreti soprannumerari), of whom there are several hundred in various parts of the Catholic world;
•the honorary chamberlains in violet;
•the honorary chamberlains extra urbem (outside the city), who are not received in their official capacity in the papal court when held at Rome;
•the official college chaplains;
•the honorary private chaplains;
•the honorary chaplains extra urbem (see 11);
•the private clerics; and
•the official college of papal chaplains.

In the case of certain of the above-mentioned classes the honorary office (together with the corresponding title and distinctive dress) lapses at the death of the pope. This is particularly true with regard to the supernumerary private and honorary Chamberlains. The reason for this is self-evident. It is possible to be prothonotary of the Holy Roman Church or cleric of the Apostolic Camera, etc.; but one cannot be chamberlain to the Holy Roman Church, but simply chamberlain to a particular pontiff, whose death dissolves the relation between the two. Unless the newly elected pontiff renews the appointment the former chamberlain returns permanently to the general ranks of the clergy. Nor is there inconsistency in the fact that certain lay chamberlains continue in the papal service immediately after a papal election. Their services are necessary to the new pontiff and he naturally recognizes such persons, which amounts practically to a tacit appointment. It is regrettable that occasionally persons thus distinguished by the pope either assume a dress arranged according to their own notions or, being dissatisfied with dress conceded, appropriate that of a higher office. The farther a country is from Rome, the more apt are such things to occur. It should be noted that members of religious orders may use the title "Monsignor" only if they are bishops or archbishops. All other ranks of the prelacy are of course closed to them, if we except the Master of the Sacred Palace, who being always a Dominican, is one of the prelates, but may not be addressed as Monsignor. The custom introduced in the sixteenth century of giving the generals of religious orders the title "Monsignor" was of short duration.

I was confused with this also. Glad you made me look it up.

I believe that prior to VC II PA's could wear a miter as well.  

That´s right, all ranks of Protonotaries had the right to Pontificals.

Some other priests of rank, like the Archpriest of St. Peter´s, the Canons of Malta or the provosts of Milan, also wore the miter.