This is less advice and more thinking through a problem. Just some quick thoughts (a rant or ramble if you will. If you have ideas on this, please share. I'd love to have a discussion on it.)
One of the things that happens in table top RPG fights is that the combat itself tends to be static. The crunchier or more tactical the RPG, the less likely you are to get characters to move. This can lead to two characters not moving at all once they get into range of each other, as instead they just trade blows and see who can deplete the HP of the others first.
This isn't a problem for the most part, but it can cause one. Static combats can quickly just become math and dice rolls. Which in turn can become completely mechanical and depending on your players that can make it more boring than exciting.
The other problem is that if you're trying for a more cinematic game, well, cinematic combats move around. And two people not moving once they get into range of each other makes them less cinematic.
So how do you fix that? It really depends on your game, and your game system. Systems that are more narratively abstract (like FFG's Star Wars, FATE, etc) you can just add it to the narrative of the fight, spending Destiny Points, Advantages, Threats, or whatever to move the fight into more or less dangerous positions.
If you're using Theater of the Mind for your combat, you can also do that - within reason - and describe how a fight moves or what happens inside a combat.
If you're using a tactical map though, it is harder. Specifically because of the tactical map positioning is important and can be the difference between a buddy being able to grenade a group of badguys safely or not. In those cases your options are more limited.
The best idea there I can think of is using a 'melee drift' roll of some sort as a mechanic. After a round without a character moving you roll a die to determine direction and the fight drifts one square/hex in that direction as the brawl goes. It's random and more fair, and gives some encouragement to move.
The other idea is to use what movement you can even then. Circle for better positioning. Move people in and out. Use options available to get around. In 5e there is no Opportunity Attack until you leave a threatened area. This lets you circle and try to find advantages that way or even break away.
With a tactical map you need to set up for it. Which in turn poses its own problem, because a moving fight on a map will quickly require a different map, or at the very least a very big map to cover the movement. Still there are options. You just have to think of what you want, and then give reasons and ways for it to work.
I feel like a dynamic environment for a combat on a grid would be a lot of fun, but also a lot of work. Which means you'd need to prep for it and then execute. Still, done right it should be a lot of fun.