Lisa Padol did run a fair number of mystery plots (we had a detective PC, which certainly helped, plus she tends to mine Call of Cthulhu and similar games for adventures) in her Kerberos Club Fate game (I played Alice), built with Fate v2 — but the rules are still fundamentally the same in Fate Core (v3) even with the reduction to the "four actions" — because the old "create an aspect" and "discover an aspect" actions are now both part of "create advantage". Avram Grumer also had some mystery plots in his (Fate Accelerated, ie v3) space game.
In short, running mystery plots traditionally using Fate works fine (as much as running mystery plots traditionally ever worked fine; as Fred Hicks says, mystery plots require that the characters find the core clues, so you need to use "succeed at a cost" if it looks like they'll miss critical clues entirely). The key is that the players ability to create background and modify the world is directly limited by the GM's power to accept or reject compels, declarations, and actions.
So the mode of play is very similar to that of GUMSHOE or Call of Cthulhu run well. The GM sets up or preps a mystery plot, and hooks the characters in (using compels if needed, but keep in mind the tightened restrictions in Fate Core that something has to go wrong (or for an event-based compel, something unfortunate has to happen) for a compel to be valid). The players have their characters make actions — rolling Overcome Obstacle to get past obstacles that would prevent them from progressing in the mystery, Declaring things about the situation that make it easier for them to find things out (keeping in mind that the GM has veto on Declaration, so this can only work if it fits with what's already going on), and making Create an Advantage attempts when they do things that will give them a lasting advantage rather than just getting someone to talk about what they saw (note that Overcome Obstacle can be rolled into a Create an Advantage roll if an action will both clear a barrier and create an aspect with a lasting effect). Eventually, they'll solve the mystery — at which point you're probably going to end up some kind of action scene (wherein they may end up using some of the aspects they created in the mystery).
This may involve the players putting together the clues you've given them and figuring out where to go next. Or it may involve the characters putting the clues together using die rolls (create advantage and overcome obstacle again, as needed — although this might be a good place to use the Brainstorming rules in Atomic Robo, or the Montage rules in the upcoming Shadow of the Century RPG). But either way, it's all pretty straightforward.
This alone is enough to run most mystery plots. Another technique you can use is the "run anything as a character" rule (although due to what skills traditionally have the attack option, I'd make sure players knew you were using this for character creation — or be more loose and narrative-focused about what skills could attack in mystery situations). With that approach, you could treat the mystery as a combat, with a failure result in the players side being a less than good result like them getting captured or derailed with a significant red herring — and knocking out the mystery meaning that they succeed in solving it. I don't necessarily recommend this approach, but it does have the advantage of measuring progress towards a goal (solving the mystery) in a very easy to measure way — without giving out the details of the mystery beforehand.
