With my L5R game, and by that I mean all 4 generations, I've ended up acquiring a lot of experience with splitting the party. With a game set around a city the PCs weren't always friends, or party members, in the classic sense and so I had to get good at balancing screen time. One of the more fun tricks I picked up, that I still like to use on occasion, is getting everyone in their individual scenarios into a fight, then running it at the same time. It's tricky, but when you do it it has some definite benefits.
Multiple Fights Pros:
The Pros for this are pretty obvious. Nothing slows down a session like a fight sequence, and little sucks more in an RPG than sitting there while other people get to play the game. So when the Rogue goes off on their own and gets into a fight, not only are you watching the rogue play while you do nothing, but it's going to take a long time. If everyone is in their own fight though, everyone is playing the game.
Another pro is that this really doesn't take more time than it does to run a big combat for the whole party. It doesn't matter if your group is split up or together in character, six players take as long as those six specific players take to do their actions.
The third pro I'll mention is that when you pull it off, you end up with a lot of neat cinematic flourishes as the camera jumps from scene to scene showcasing the different fights going on. It can give a real feel of watching a movie or show as you see Bob's critical strike in scene 1, then jump to Sarah getting wrecked in scene 2, and then Alex preparing a powerful spell in scene 3.
The cons is really just the con, and it can be a doozy: people can get confused. RPGs and theater of the mind's eye already has problems where people mix up what happens. Having 3 fights going on at the same time can make that even worse.
You as the GM need to have a clear understanding and the ability to hold the multiple scenes completely separate in your head. When Sarah asks what is going on, you can't tell her what is happening with Alex or Bob. ANd if Sarah isn't sure what is going on, and isn't plainly aware that Alex and Bob are nowhere near her, she may wonder why she can't attack the guy who Bob just injured to take him out of the fight.
As the GM this means a lot more fiddly bits to keep under control in the fight, and you need to be prepared to handle this. You need the enemy stat blocks separated in some way, but just as accessible. You also need to be ready to jump from scene to scene as turns come up.
How You Do It
Execution, luckily, is the easiest part of this provided you're prepared. Have everyone roll initiative. Roll initiative for the opposition as normal. Then just handle initiative as normal. As a character's turn comes up, you bring the focus to that scene. When the turn ends and someone else's turn comes up, you swap to that scene. If they're both in the same fight, great. If not? You switch.
That is really all there is to it.
Well, there is one other thing:
Remember, multiple fights means none of them have the full party
Multiple fights is a great way to wipe the whole group. Why? Because a lot of GMs prepare combats for the whole group, and in this situation you don't have the whole group in any one scene. If the PCs are getting themselves killed that may be on them, but just make sure that if that isn't the idea that the fights are scaled appropriately.