2009- Leauge of legends launched and you saw other Moba style games compete with it. Most notably DoTA and to a lesser extent HotS.
This lasted for a while and then we got the evolution of the moba that I think began with smite (2014) which featured a new perspective on the moba but ultimately ended with Overwatch being the' class & team-based FPS in 2016. Between the release of these two games you had lots of games that utilized concepts that originated in TF2 but were expanded on and experimented with. Think- "payload" and "checkpoint" style maps respectively.
There also seemed to be an interesting trend of team-based FPS titles such as Vermintide and Rainbow six siege but that's more of an offshoot I think.
Now the most popular genre (maybe not in our circles) is the battle royal. PUBG probably is the first game that comes to mind but it wasn't the necessarily the first battle royale game so much as it was the first to combine a number of mechanics all into one game. If you go all the way back you can trace this game's mechanical structure to DayZ just about. Then, you had a myriad of copycats that wanted to be "the next DayZ". Some doing better (7 days 2 die & Ark survival evolved) than others (InfestationZ) What you saw with these titles were scavenging and base building intermingled with gunplay.
I don't know who did it first but someone got the idea to do the battle royale scenario which was almost directly stolen from Battle Royale manga/anime. See also the hunger games if you're a fucking numale.
Now then comes July 25, 2017 the release of Fortnite. What do you get when you streamline that earlier mentioned: "scavenging and base building intermingled with gunplay"? You get fortnite (and also get their battle royale mode that comes with the stand alone game). That's pretty much where we are right now.
The base game of fortnite is not a battle royale game though. I'm still trying to figure out how to place it since it blends so much shit together it's hard for me definitely determine what it is. Whatever it is, I've found pretty addicting.
tl;dr
So we've gone like this
>Moba
>then FPS moba
>class and team based FPS
>transition to survival FPS
>prominence of the battle royale genre emerges
>now we're here.