Been playing Civ since Civ 1. Unfortunately, 6 was a bit of a miss in my opinion. Civ 5 is still as addictive as nicotine.
On another side, the multiplayer in Civ 5 was always broken, the saves didn't load, someone getting dropped almost always meant 2+ turns lost for them. Hopefull this one doesn't suffer from this any more.
It introduced interesting new systems, but the agenda system usually feels more annoying than actually offering interesting long term strategic options. Same goes for religion, the religious combat mechanic is too fiddly, leading to me mostly just ignoring it.
Civ 6 is a deeper game, but for whatever reason I can't put my finger on, it was just a bit less fun.
In VI the Barbarian is faster than you, so it gets out of sight quick, and from then on you get one wave of Barbarians after another.
Also, I didn't understand the mechanics of the new religion system, but I think I got it on a basic level at least by now.
In some ways, I like VI (the district system can be infuriating, but is basically a good idea), I like the graphics, as well, but I never got "into" it as I did with V.
The way it the barbarians work though is somewhat clever: they send out scouts, and it seems like until their scout spots your city they won't spawn attackers. So you're incentivized to harass their scouts early on at least, but it's not really explained in any fashion beyond perhaps a visual ding.
After that the AI is so bad that you've basically won once the barbs are no longer threatening. I got back into civ 6 last month after a two year break and had forgotten this dynamic and kept starting over when the barbs felt overwhelming and killed a slinger or captured a cheeky unprotected settler.
You also may be moving in a straight line over bad terrain, while the AI is making sure it is always using the flattest terrain to get the most movement per turn.
TL;DR - The issue is unit specific. Build units that have more movement per turn, and have few penalties moving over bad terrain, and Barbs won't get away from you.
Also, just a tip; Don't chase them. Focus on finding their camp, because thats where they spawn. If you start attacking that, Barbs will naturally come to defend it. Spend your movement and units to find and destroy the camp. Far more valuable than trying to eliminate specific Barb units. If left unchecked, the camp will spawn horseman or spearman which are much harder to deal with. It's way better to have a wandering scout on the map with no more barb spawn than hunting down a few scouts and leaving the camp in tact.