What was their business in Ferrara in "My Last Duchess"?
Towards the end of the monologue the Duke says, "We'll meet the company below, then." Evidently the Count and his daughter are present at the Duke's palace, along with some others who may be relatives and friends. The Duke has withdrawn upstairs to a private room ostensibly to show the Count's representative some of his art collection but really to discuss the matter of the dowry the Count will give him along with his daughter, who may be only a girl of around fourteen like the Count's former bride. We know they have come upstairs to discuss the dowry because the Duke says, "I repeat, / the Count your master's known munificence / Is ample warrant that no just pretence / Of mine for dowry will be disallowed." He has already said this once, and now he is repeating it. That is what is primarily on his mind, and that is why he has brought the Count's representative up to this secluded room. Showing the artwork is only an excuse. We can be quite sure that the Duke will be requesting a very large dowry.
There are some suggestions implicit in the monologue that the Duke is capable of having wives murdered if they dissatisfy him in any way. A girl who brought too small a dowry might dissatisfy him. By showing the Count's representative his art collection and mentioning the famous names of the artists, the Duke may further be hinting that he is a very wealthy man with very expensive tastes and therefore would expect a bride to bring him a dowry commensurate with his own value as a husband.
The Duke is formally receiving an ambassador from a Count, who wants to offer his daughter as the next duchess, in exchange for a handsome dowry. The host Duke, in a conceited, ironic tour of his house, is trying to impress the ambassador with his prestige and his taste. In the process, he actually reveals his avarice, his cruelty, and his overweening pride. The ambassador’s task, besides formally suggesting the liaison, is to assess the Duke’s character, that is, whether he would be a good match for the daughter. By the ironic tone in the monologue, Browning manages to tell the reader that the ambassador sees through the Duke’s character and will bring a negative report back home; this match will never be made.