Let me put it like this. Here's a Math Problem for you to solve:
I'm sitting down with a group of 4 players and we're playing D&D. The players are exploring a dungeon. They decide to split up to cover more ground. Players 1 and 2 go off together. Players 3 and 4 go off together.
Suddenly, Players 3 and 4 enter a chamber and get into a fight with a hive of spiders. Players 1 and 2 are approximately 600 feet away in a totally different area of the dungeon. If Players 1 and 2 can only run 60 feet per round, how many rounds will it take them to get to their party members 3 and 4?
Pst. Answer: 10 rounds.
Can I tell you? This is a moment for excitement. Players 3 and 4 are stuck in a huge fight with a hive of spiders. They're running in the direction of Players 1 and 2, and they are running round after round for their lives while a host of spiders chases them. Meanwhile, Players 1 and 2 are desperately trying to regroup with them round after round to try to back them up. If both groups run towards one another, they could reach one another in 5 rounds. Each round is blood pumping for at any point in time more enemies could pop out at either group to add more enemies to the mix. Also, if even one of the players, 3 or 4, rolled low on initiative, it may be that spiders get to run up to them and attack each round, and they either have to Disengage each round to avoid AOO or they have to risk AOO and Dash each round (because the spiders can also Dash and catch up each round). Will Players 1 and 2 reach them in time?
This is how splitting up should work. It shouldn't be that Players 3 and 4 get into a huge fight and players 1 and 2 can sell equipment, take a nap, have some convos, explore the rest of the dungeon, steal everything from enemies, etc. HECK no! They should be thrown into turn-based combat as well, and they should come running to the aid of their party members 1 round at a time so that they don't just randomly show up after 1 round of combat and partake in it when they were 600 feet away. The whole point of scenarios like that are that it is a race to try to go save your friends before they get caught and die.
I can't tell you how many exciting sessions I've played out with scenarios like this. It's like, "Holy Shoot! Jon and Anne are in serious Poop! We can't get to them fast enough. Do we have any haste potions we could use or spells to get us there faster? ANYTHING!!!"
I'm just saying, the world should freeze. If someone's in a conversation with someone else, then yes, their conversation should finish and then they're thrown into combat, but otherwise, the world should pause and people should be in Turn-Based mode. The only exempt thing that I can think of would be dialogue, and even when it is finished, they shouldn't just teleport to the battlefield. They should just pop into Turn-Based mode immediately and have to move and do round after round in initiative order like anyone else. So if they were trading with Arun at the Grove and their party is in the Goblin Camp, they should have to spend round after round running to get to the nearest Waypoint so they can fast travel to the Goblin Camp Waypoint and then run turn after turn to where their companions are.