I'll talk a bit about the slice of the question that I know well enough to confidently speak on: Charlemagne and Alcuin of York.

In every language pronunciation changes over time. One of the reasons that English language's spelling is so unphonetic is that in many cases English spelling records how words USED to be pronounced even if the pronunciation shifted away hundreds and hundreds of years ago. For example the "gh" sound in words like "knight" used to have the kind of back of the throat sound that doesn't really exist in English anymore but we still spell it that way even though English hasn't had that sound for a very long time.

The same thing happened fairly early on with Latin but it's really hard to detect exactly how and when it happened since people kept on using old Latin spellings. So just like with "knight" people would keep on spelling it the old way and then pronounce it in the way everyone actually said it. However since we don't have audio recordings of how people were actually talking it can be really hard to nail down how people's pronunciation was drifting away from their spelling. We can approximate by looking at how people misspelled some words and by extrapolating back from later Romance languages but the whole thing is pretty murky and there's a lot of areas without a solid scholarly consensus.

The main upshot of this is in the early Middle Ages people in church would probably not have a hard time following what the priest was saying since even if the priest was using proper Latin grammar that their local dialect didn't much use the priest's pronunciation would be pretty close to what they were used to so they'd be able to follow along.

This didn't last forever though, eventually the gap between the Latin priests were writing and what people were saying became harder to bridge. The big name here is Alcuin of York, specifically what he did in the court of Charlemagne. Alcuin obviously came from York and was thus not a native speaker of a Romance language but rather learned it out of books so he pronounced things closer to how they were spelled because that's what he knew. This wasn't just him but the Northumbrian (loosely modern northern England) scholarly community in general and back from there to the Irish monks (who also, obviously, were not native speakers of Romance languages either) who had an enormous influence on the early church history of Northumbria.

So Charlemagne hires Alcuin and his various sidekicks to work at his court in Aachen and run his palace school (Urbs Regale) where the princes etc. were taught. So you have Alcuin coming in with a big reputation from Northumbria which well known as a center of scholarship (before the Vikings of course) speaking Latin more like it was spelled instead of using the proto-French pronunciations that most people there were using.

Since Alcuin was an influential guy, with a very important job, and sponsored by a very powerful ruler a lot of people followed his lead and began speaking in what Alcuin understood to be proper Latin. Of course just one person can't reshape history all by himself and a lot of other people were involved but Alcuin is the single biggest name.

So what you had happening is that Alcuin (and people influenced by him) were trying to drag Latin back to classical Latin (of course what Alcuin thought was proper Latin and how actual classical Romans spoke were not always the exact same thing, but he was trying) but random people in the countryside weren't about to change their whole way of talking just because Alcuin said so their spoken language kept on diverging from Latin into the various Romance languages. The result of this is that you started getting more and more of a split between Church Latin and the language everyday people spoke.

So if you think about it in the right away and squint a bit, French exists because of an English guy.

Editing in a few more ideas:

At this time it's hard to talk about discrete dialects since there would be a broad spectrum of different dialects. For example if you walked all around what's now Spain, Portugal, France, and Italy and avoided non-Romance speakers like Basques and Bretons then in basically every case each town would be perfectly able to understand the next town over and there wouldn't be much change at all along borders. But as you travelled gradually the differences would stack up and you'd have a harder and hard time being understood. Of course there were broad regional dialects but the borders between them were quite fuzzy.

When people had a hard time understanding each other one of the things they did is make up a whole new language: Sabir/Mediterranean Lingua Franca which was the closest thing to D&D's "common" language. Basically start with Northern Italian dialects, radically simplify the grammar, then add in gobs of vocabulary from southern French dialects, Catalan, Greek, Arabic, and Turkish. It wasn't written down much but it was used a lot by sailors. Here are some examples of it: https://minds.wisconsin.edu/bitstream/item/3920/edition3/texts.html