Fringe started as a replacement for The X-Files and a huge television event, everyone thought iw as going to be the next 'Lost'... it wasn't, but it became it's own things and while the first few seasons heavily depicted Fringe events as a case of the week, the show became more and more serialized as it moved forward. Parallel universes, clones, time travel and more made the final season one long story about them changing the future to save the past.
The show was actually very spooky at the beginning, and even had this desaturated sepia look that they sorta dropped. But it tackled mostly monster of the week through mystic folklore, and while it had a small serialized element with them looking for their missing father, the show took a turn around season 4 when the angels came into the picture. We're now entering the final season and while the cases of the week remain every now and then, most of the show is about how they have to stop the apocalypse created by god while they have to save Lucifer's son that sorta became their little brother.
It was pretty easy and fun to watch. He got the thingy in his head and now he has flashes. It was ok and it was mostly procedural for a good while, until he got new abilites with the thingy 2.0 in his braind and the show became more serialized delving into Chuck's family relationships and his love for Sara.
The string of case-of-the-week episodes after the pilot were tough to get through but by the end of the first season the show really starts to find its footing, and like most of the ones in this list, it became better with it.
4. Marvel's Agents of SHIELD
The first few episodes tried to tell procedural stories about how the Agents solved MCU related cases. Everything changed when Captain America: The Winter Soldier premiered, revealing Hydra was SHIELD and the show had to change format to them being on the run and rebuilding SHIELD. Every season after that has been one long (or short) serialized story after the other.
This had to be the procedural with the best cinematography i have ever seen. The first episodes start out as procedurals, however it's clear it's leading to something early on. The procedural stories also often tie in with the serialized story thematically. In the early episodes you could see the 'case of the week' stories as a stepping stone or foundation for the main story, by season 2, the show became deeply serialized about the realtionship between the leading
Joss Whedon's Buffy wannabe had a nice premise: We use people as empty dolls for a few years, and we wiped their minds everyday to give them different abilities and we basically rent them to millionaires. The first few episodes were very procedural-ish showing us how the millionaires used the dolls for their crazy weird fetiches, but halfway through season 1, the show took a turn for the better when a resistance was revealed, by season 2, the show time jumped to an apocalypse in the future.
The doctor who spin-off started as the X-files meets Silk Stalkings, set in the Whoverse. The first few seasons were very procedural, with a few serialized elements here and there, by season 3 the show changed formats to miniseries and told one single story for season 3 (all the children in the world stopped) and another one for season 4 (nobody in the world could die).
Source: Me, myself and I.
ONTD, any procedural show that became serialized that you like?