I haven't done barrel fermenting but it's quite common in Burgundy with Chardonnay. Usually, its done in neutral barrels. The reason it's done is that it adds complexity and mouthfeel to the finished product because of the interaction with oak. The process is to crush, press and rack into barrels and pitch the yeast and ferment in the barrel. No skins involved.
Even though I have made barrel aged Chardonnays, I have never barrel fermented. I always fermented in stainless tanks and then moved to barrels. First, it's hard to see how full the barrel should be and sometimes the fermentation throws a lot of foam. Unless you have a lot of head space, you can't just slap an airlock on in a bung on there. I've seen guys just loosely put a bung on the barrel and let it foam all over the winery floor.
Many people conduction MLF at the same time as yeast fermentation or shortly afterwards when there is no sulfites present.
After it's done, you will need to sacrifice the wine in some barrels to top up in the other ones if you want to continue to age in barrel and on the lees. If you just want to age on the lees, I would just put in a stainless tank.
There is a tiny bit of color picked up from the barrel fermentation process. Kind of depends on how much of the char is left in the barrel.
Just google barrel fermented chardonnay and you'll get more results than you could possibly look at.