This presumes, of course, that one of the parties is a Catholic. It is commonly called a convalidation, although that term is a little off the mark (but it still works for common purposes). Google the term and you will find resources.
This would require the same pre nuptial investigation needed for any Catholic celebration of marriage to determine the freedom of each party to marry. Generally any other aspect of pre nuptial preparation that is required for any Catholic wedding is also required. I agree that the reasons why a Catholic did not marry validly and properly the first time need to be disussed with the pastor. That discussion though takes place in the context of a “welcome back home.”
Such a “blessing” is in reality not a blessing but the creation of a valid marriage where one had not existed previously. It needs to be created in the same way as any marriage involving a Catholic — by the expression of true marital consent by parties who are free to marry with the assistance of an authorized priest or deaon.
This requires some recognition by both parties that the prior ceremony, even though they intended and expressed consent to marry, still did not result in a valid marriage in either the sight of God or the Church. At the second proper celebration of marriage, they must place true consent to marriage.
The diocese is involved in the sense that its bishop may have established policies for the matter. If dispensations or permissions are needed for a valid and lawful marriage, the diocese would be involved.