The woman and the kid Vaughn was consoling was the family of one of his thugs, Stan, they found dead in a warehouse or something a few episodes ago, supposedly killed by Mexicans.
Vaughn was looking for the girl that pawned the stuff from Caspere's house, Caspere being the guy murdered in the first episode, with whom that whole investigation started, he was the city manager and he had Vaughn's money when he died. Vaughn is looking for a disc that belonged to Caspere, with recordings of a lot of higher-ups involved in orgies and whatnot. He was promised a deal on the railroad (even though his money was lost with Caspere's death) if he can retrieve it.
The girl they rescued was the missing sister of a Mexican (I think?) lady they met during their investigation in the first episode. The lady cop was investigating, that woman told her her sister's been missing and she promised to find her.
Those parties are crucial to the plot, with all the richfolk involved and to be fair I'm kinda confused about this by now. It's something about those diamonds they are looking for but I'm no longer certain what that's about, someone else should probably explain this better.