The overarching plot is dealt with in the first, fourth and fifth cases, while the middle two ("Recipe for Turnabout" and "The Stolen Turnabout") seemed irrelevant. However, they foreshadowed major events in the metaplot. In Case 2, the true killer is Luke Atmey, a man whose name alone shows what he wants. He's an incredibly arrogant criminal who wants to be the centre of attention and commits crimes to cover up his other crimes. Just like Dahlia Hawthorne. You could also read Ron DeLite as being a representation of young Phoenix from case 1, or Terry Fawles as all were wrongly accused and victims of the plan of the real killer.
In the third case the victim's coffee cup is poisoned as he manages to finally escape his debt with Furio Tigre. Like Diego Armando was poisoned by Dahlia Hawthorne just as he looked set to get case breaking evidence to implicate her for the murder of her sister, the theft of the diamond, creating a false identity, causing the death of Terry Fawles and tampering with evidence. Those aren't the only parallels that could be found.
The Très Bien case contains a fake murder in which a witness who is sex-minded and older than most Ace Attorney characters says that they saw the actual murder. This is similar to Bikini and the removal of the Shichishito from Elise' body.
Furio Tigre looks like Phoenix. In this case Furio is the murderer and even parallels Dahlia's poisoning of Diego. To me this makes sense as to Godot, Phoenix is the villain who got Mia killed and Furio reflects that.
There's a triple parallel in both cases with the theme of false identities and impersonation. Much like with Tigre/Nick and Iris/Dahlia, a major breakthrough came when it was realized that there were actually two Mask*DeMasques running around, Luke and Ron. As well, the ridiculous efforts of Ron to keep his wife happy and the awkward romantic fumblings of Gumshoe for Maggey are writ large with Godot's attempts to protect Maya and destroy Dahlia for Mia's sake.
The Ron/Desiree thing resembles the relationship between Terry Fawles and Dahlia Hawthorne a little bit. Everything that was problematic between Ron and Desiree is taken Up to Eleven with the nightmare that is Terry x Dahlia.
Viola actually admits that what Furio was doing to collect on his debts was "evil," but that she was helping him with it because he pretends to care about her and she wants to believe that he does. This reflects things in the main story.
Also, 3-3 features a criminal who pretends to be someone way more trustworthy than themselves in order to frame an innocent party. Just like Dahlia pretending to be Iris in the last case — only with Dahlia, the framing isn't as important as the crime itself. (Also, with Dahlia, the player doesn't see it coming from a mile away.) Since Tigre turns out to be a bit of a coward (trying to feebly persuade the court that 'it was probably someone else' when everything's against him), the contrast is perfect — Dahlia puts up an innocent front, but is a heartless monster who will do anything to hurt those she wants to hurt, stating that she doesn't care because she's already dead. Tigre puts up an aggressive, intimidating front, but is very cautious to make sure blame doesn't get pinned on him, even recreating the entirety of the murder of Glen Elg with changed guilty parties.